Oil pump, or blocked oil lines. The clicking is the valve lifters not getting oil. You might try running some Sea Foam in it, follow the instructions carefully.
for concerns like this take the car and the old battery to an auto electrician
have the old battery load tested
then have the charge system properly tested
That includes the alternator , drive belt condition , voltage regulator
Un-do the hose clamp on the air box and the one on the throttle body, carefully disconnect the MAS sensor plug at the top of the hose... remove the hose and inspect it for cracking or breaks. Very common... This condition causes un-metered air to enter the system. The new part is about $80.00 on ebay. It will likely come with a new MAS sensor. JUST BE SURE AND DOUBLE CHECK THE AIR FLOW DIRECTION arrow on the sensor. The last one I did, the part was good but the sensor was installed backward. Took me a while to find that problem.
Sounds like the wheel nuts are not fully tightened on the wheels.I would check that first.Some shops use a "torque" bar when tightening wheels which can result in loose wheel nuts due to rusty/tight/damaged wheel stud threads creating a false torque reading and loose wheel nuts.
If the starter is pulling to many amps to crank the engine,it will drop the battery voltage too low for the fuel pump to operate at all or operate properly and the car will not start if it is not getting fuel at the proper pressure or no pressure.
Usually an indication of a bad ignition module. Depends on which engine you have, but as the module gets warmed up, it will get an open circuit inside of it. As it cools off and shrinks back down, it will start up an run awhile. Hope this helps
Well that's where doing some testing comes into play . Instead of asking on here an expecting a magical fix , learn how to test an lean about the different systems on your vehicle . Test the fuel system , fuel pump pressure an flow , testing electrical circuits with a DVOM - digital volt ohm meter . Eliminate thing's that are right . Find service information manual so you know what the fuel pressure is suppose to be , how much current the fuel pump is expected to draw . Also the electronic ignition system specification. I think it's cool you want to find out about you vehicle , but go to school an learn about it the right way . Because listening to someone guess will get you no where . Your vehicle is a rolling computer , has a lot of electronics . Electric Testing Techniques You Need to KnowMotor Age TST New Battery Starting Charging System Testing TechniquesFuel System Diagnosis Presented by Delphi
If you don't understand any of this , do yourself a favor an take it to a qualified repair shop . Repair modern automobiles isn't easy , even for seasoned Technicians .