How do I detect the problem for a clicking sound as I start the engine?

Bit of a Mess , get Workshop or Garage to look at your faults

Sounds like a vacuum line got knock off somewhere causing a vacuum leak and the high idle.

It could be several things. The fan relay may be bad, The fuse for the temp. gauge could be blown, or the temp sensor could be bad.

Almost sounds like an injector heat-soak issue... However I'm wired on coffee and Monster at the moment. By not starting for a few days, do you try to start it several times in the interim, or do you just not touch it out of frustration for a few days? You should be able to check for heat-soak the next time it won't start after a drive by pouring a bottle of cool water over the injectors and down the fuel rail. Should cool off the injectors so they work properly again.

Have you checked to see if there are any codes coming from computer? With the key in the ignition, turn it from off to on three times (not all the way to start, ie crank the motor). So you go on-off-on-off-on-off then back to on and leave it there. The check engine light will flash a specific sequence. It may be handy to have a piece of paper in case there's multiple codes given. Basically what you do is count the flashes. They'll come in two-digit sequences, with the digits separated by a small pause. If there are multiple codes, they will be separated by a longer pause. So flash-flash-pause-flash-flash-flash paauuusssee flash-flash-flash-pause-flash-flash-flash would be a 23 and 33 respectively, if that makes sense.

This may help you with trouble-shooting, especially if the computer does give you some codes. Don't worry about Code 12 and Code 55, as 12 is battery/power disconnected within last 50 engine starts (or something like that), and 55 is end of message (ie, no more codes).

Edit**
Codes for your year are:
Code 88-Display used for start of test.
Code 11-Camshaft signal or Ignition signal-no reference signal detected during engine cranking.
Code 12-Memory to controller has been cleared within 50-l00 engine starts.
Code 13-MAP sensor pneumatic signal-no variation in MAP sensor signal is detected or no difference is recognized between the engine MAP reading and the stored barometric pressure reading..
Code 14-MAP voltage too high or too low.
Code 15-Vehicle speed sensor signal-no distance sensor signal detected during road load conditions.
Code 16-Knock sensor circuit-Open or short has been detected in the knock sensor circuit.
Code 16-Battery input sensor-battery voltage sensor input below 4 volts with engine running.
Code 17-Low engine temperature-engine coolant temperature remains below normal operating temperature during vehicle travel; possible thermostat problem.
Code 21-Oxygen sensor signal-neither rich or lean condition is detected from the oxygen sensor input.
Code 22-Coolant voltage low-coolant temperature sensor input below the minimum acceptable voltage/Coolant voltage high-coolant temperature sensor input above the maximum acceptable voltage.
Code 23-Air Charge or Throttle Body temperature voltage HIGH/LOW-charge air temperature sensor input is above or below the acceptable voltage limits.
Code 24-Throttle Position sensor voltage high or low.
Code 25-Automatic Idle Speed (AIS) motor driver circuit-short or open detected in 1 or more of the AIS control circuits.
Code 26-Injectors No. 1, 2, or 3 peak current not reached, high resistance in circuit.
Code 27 -Injector control circuit-bank output driver stage does not respond properly to the control signal.
Code 27-Injectors No. 1, 2, or 3 control circuit and peak current not reached.
Code 31-Purge solenoid circuit-open or short detected in the purge solenoid circuit.
Code 32-Exhaust Gas Recirculation (EGR) solenoid circuit-open or short detected in the EGR solenoid circuit/EGR system failure-required change in fuel/air ratio not detected during diagnostic test.
Code 33-Air conditioner clutch relay circuit-open or short detected in the air conditioner clutch relay circuit. If vehicle doesn't have air conditioning ignore this code.
Code 34-Speed control servo solenoids or MUX speed control circuit HIGH/LOW-open or short detected in the vacuum or vent solenoid circuits or speed control switch input above or below allowable voltage.
Code 35-Radiator fan control relay circuit-open or short detected in the radiator fan relay circuit.
Code 35-Idle switch shorted-switch input shorted to ground-some 1993 vehicles.
Code 37-Part Throttle Unlock (PTU) circuit for torque converter clutch-open or short detected in the torque converter part throttle unlock solenoid circuit.
Code 37-Baro Reed Solenoid-solenoid does not turn off when it should.
Code 37-Shift indicator circuit (manual transaxle).
Code 41-Charging system circuit-output driver stage for generator field does not respond properly to the voltage regulator control signal.
Code 42-Fuel pump or no Autoshut-down (ASD) relay voltage sense at controller.
Code 43-Ignition control circuit-peak primary circuit current not respond properly with maximum dwell time.
Code 43-Ignition coil #1, 2, or 3 primary circuits-peak primary was not achieved within the maximum allowable dwell time.
Code 44-Battery temperature voltage-problem exists in the PCM battery temperature circuit or there is an open or short in the engine coolant temperature circuit.
Code 44-Fused J2 circuit in not present in the logic board; used on the single engine module controller system.
Code 44-Overdrive solenoid circuit-open or short in overdrive solenoid circuit.
Code 46-Battery voltage too high-battery voltage sense input above target charging voltage during engine operation.
Code 47-Battery voltage too low-battery voltage sense input below target charging voltage.
Code 51-Air/fuel at limit-oxygen sensor signal input indicates LEAN air/fuel ratio condition during engine operation.
Code 52-Air/fuel at limit-oxygen sensor signal input indicates RICH air/fuel ratio condition during engine operation.
Code 53-Internal controller failure-internal engine controller fault condition detected during self test.
Code 54-Camshaft or (distributor sync.) reference circuit-No camshaft position sensor signal detected during engine rotation.
Code 55-End of message.
Code 61-Baro read solenoid-open or short detected in the baro read solenoid circuit.
Code 62-EMR mileage not stored-unsuccessful attempt to update EMR mileage in the controller EEPROM.
Code 63-EEPROM write denied-unsuccessful attempt to write to an EEPROM location by the controller.
Code 64-Flex fuel sensor- Flex fuel sensor signal out of range-(new in 1993)-CNG Temperature voltage out of range-CN gas pressure out of range.
Code 66-No CCD messages or no BODY CCD messages or no EATX CCD messages-messages from the CCD bus or the BODY CCD or the EATX CCD were not received by the PCM.
Code 76-Ballast bypass relay-open or short in fuel pump relay circuit.
Code 77-Speed control relay-an open or short has been detected in the speed control relay.
Code 88-Display used for start of test.
Code Error-Fault code error-Unrecognized fault ID received by DRBII.

Please let me know.

Yes here is that list.
Chrysler products COMMON CODES OBD1 (vehicles made before 1995)
Part 1: Chrysler Domestic vehicles
CODES/Meaning
88 display used for start or test
11 camshaft signal or ignition signal, no reference signal during cranking
12 memory to controller has been cleared within 50-100 engine starts
13 map sensor not detecting any change during test
14 map voltage too high or too low
15 vehicle speed sensor, no signal detected
16 knock sensor circuit. open or short detected
17 low engine temperature, possible thermostat fault
21 oxygen sensor signal, neither rich or lean detected
22 coolant sensor voltage low
23 air charge voltage high/low detected
24 throttle positioner sensor voltage high/low
25 automatic idle speed motor driver circuit, short or open detected
26 injectors 1,2,3 peak current not reached
27 injector control circuit does not respond to control signal
31 purge solenoid circuit, open or short detected
32 egr solenoid circuit, open or short detected
33 a/c clutch relay circuit, open or short detected
34 speed control servo solenoid, open or short detected
35 radiator fan control relay circuit, open or short detected
36 wastegate solenoid, open or short detected
37 part throttle unlock(PTU) circuit, open or short detected
41 charging system circuit not responding to control signal
42 fuel pump or auto shutdown (ASD) relay voltage sensed at controller
43 ignition control circuit not responding
44 battery temperature voltage circuit problem
45 turbo boost limit exceeded-map sensor detects overboost
46 battery voltage too high
47 battery voltage too low
51 air/fuel at limit
52 logic module fault
53 internal controller failure
54 camshaft reference circuit not detected
55 end of message
61 baro read solenoid, open or short detected
62 emr mileage not stored
63 eeprom write denied
64 flex fuel sensor signal out of range
65 manifold tuning valve, open or short detected
66 no ccd messages
76 ballast bypass relay, open or short detected
77 speed control relay, open or short detected

Yes the starter is going outreplace asap. it has a short that is only there when it is hot. things expands in the starter

I wonder if your battery is really in an excellent shape. The way you described seem to be non sufficient voltage for the starter solenoid to engage completely. Make a quick observaton by leaving the headlight on while cranking the engine. If no significant drop on the headlight brightness. Then your battery is assumed to be good. (a cold crank voltage shouldn't below 10V) Then the remained items to work with is either the solenoid or the starter or both. I believe the solenoid is piggy pack on the starter. Most autopart stores would test it for free of charge.

replace the fuel filter

sounds like low fuel pressure have it ckecked

sounds like your starter motor might be stuffed....might need to replace it