What color are the ignition sense and crank wires on a 09 chevy cobalt

it's a manual means my transmission is manual transmission?
with a left foot clutch ,the clutch switch is bad above the pedal clutch
that switch enables the starter relay just like all cars, manual
so you an not start engine in gear.

those other problems are other causes.
IMMO key must not be ignored. padlock
thee users guide tells you what the lamp does. (lost that?)
lost the second ignition key? from when car was new.
some cars dim HL lamps to have extra power to crank the engine, a super good features on all cars with it.
you just buy a dead car, or did it just fail out of the blue.??????????
learn what RAP is in the manual.?
key off radio works for delay10mn them or ends door open
page 2-18 in the manual

now IMMO words same manual free, to all chev. owners.
Page 2-15 actual
to start the engine closes door or IMMO Passlock
will fail.
Your vehicle is equipped with the Passlock®
theft-deterrent system.
Passlock®
is a passive theft-deterrent system. Passlock®
enables fuel if the ignition lock cylinder is turned with
a valid key. If a correct key is not used or the ignition
lock cylinder is tampered with, fuel is disabled
During normal operation, the security light will go off
approximately three seconds after the key is turned to
RUN following an engine start.
If the engine stalls and the security light flashes, wait
until the light stops flashing before trying to restart
the engine. Remember to release the key from START
as soon as the engine starts.
If the engine is running and the security light comes on,
you will be able to restart the engine if you turn the
engine off. However, your Passlock®system is not working properly and must be serviced by yourdealer.

READ
https://www.manualowl.com/am/Chevrolet/2005-Cobalt/Manual/388


but the clutch switch is failing and can be jumpered easy, your paper clip. some switches have adjustment or are just fell loose.

if only clicks test starter and ignition system --and recheck wiring /connections from doing work ---check engine by turning engine by hand make sure not binding or seized

check and test battery -fuses-relay --check key for wear same with ignition cylinder-- recheck install of starter

May need to check fuses and relays

Hie its the ignition key its not giving feed to the starter relay

Before you do any thing check for loose or bad connections. Also check you ground connection to engine. Check Fuses, relay and for a Fusible Link open circuit. Now let me explain what a Fusible Link does. They are short lengths of small gauge wire installed between the Battery and the Ignition Switch, or the Battery and the Fuse Panel. They'll melt and open the circuit before damage occurs in the rest of the circuit. I hope this was helpful you.

hook up a to the obd and comand the ecm to prime the fuel pump if it doesnt prime its the ecm providing wires and relays are good

Park your Chevy Cobalt on a level surface, activate the emergency brake and place blocks behind the back tires for safety. Disconnect the negative battery cable with a terminal puller. Raise the front end of the car with a jack for easier access to the starter.
Remove the intercooler pump's outer bracket in models with a 2.0L engine. Place the bolt, and all other nuts and bolts, into the jar for safekeeping until later.
Disconnect the starter electrical connections, including the S-terminal and positive battery cable. Label the wires if you have concern about remembering where they go when you install the new starter.
Unscrew the starter mounting bolts and remove the starter from the vehicle's engine.
Take your malfunctioning starter to the parts store with you. Aftermarket distributors remanufacture most of the starters they sell, so you can likely obtain a partial credit toward your new starter by exchanging your old one. Be sure that you take care not to damage the starter in transit, because the better condition it is in, the more you can get for it.
Purchase a replacement starter that has the same or higher cranking capacity (power rating) as the original starter. Also, match the bolt patterns, drive gears and electrical connections. Your owner's manual may list specifications or the auto parts technician will help you determine them. Download a free owner's manual by registering with the Chevy Owner Center (see Resources).
Install the new starter by reversing the removal process. Torque the starter mounting bolts to 37 foot pounds (50 Nm). Tighten the bolt for the intercooler pump bracket to 15 foot pounds (20 Nm).
Lower your Cobalt back to the ground. Reconnect the negative battery cable. Turn the key in the ignition to see if your new starter motor functions correctly.

yellow. you can splice into your factory wiring and work the same as if your ran one straight from the battery

The remote start should come with instructions, you will need these especially for the diagram of the units wiring harness.  It will need to be wired into the ignition key on wire, the ignition starter wire, power and ground, and usually the brake light wire, and a hoodpin disable switch that will come with the unit.  Also most manufactures will have a website with vehicle specific details like color and location of each wire on the vehicle.  hope this helps.