The door speakers in my 2008 trailblazer produce little to no sound.

The door speakers need to be replaced.

Check speaker, sub another speaker to check wiring or use analog volt meter, if no action on meter or sub doesn't work either then check wiring between door and chassis. Some wires may break inside due to continuous door use, most common on drivers door. You can try a light pull on the two wires here after sliding boot to one side, easier said than done. Lastly radio output transistor is bad, I lost one at a time until radio had no sound.

I would check your water pump. Sounds like that is going out. Pretty easy fix.

mazda 6 2008 I also have 2008 Chrysler Pacifica and I had purchased its manual from online store reliable-store.com last week. My car is fine as new now. Didin't need any mechanic or guidance. . Here's the site link
I am sure they will help you

check for a bad earth in the system . Possibly the earth strap on the motor or dirty battery terminals . IF the temp gauge will not work (bad earth ) and the ac stops working ( cabin air temp reaches setting) or the circuit has an earth problem switching off the ac. Check all wires for broken , not connected properly and check each earth point for clean and tight.

I have a 2008 Silverado and noticed my chime was not working. They work off the front door speakers. Found out GM put the cheapest speakers on the market in them. Bought a new pair of Kenwood speakers from Crutchfield and installed them last week. They sound great. I will probably install Kenwood's in the rear doors since it's only a matter of time before they die too.

Could have damaged the speakers or the radio or both.
You could pull a door panel and connect a small used speaker to see if the radio is producing a feed to the door.

In my 2008 Cobalt, the front speaker in the driver door went inoperative - no turn signal noise, no warning chimes for door open with key in, seatbelt chime, low fuel chime, and of course, no music. All of those sounds come through the speaker. I switched the driver door speaker and the passenger door speaker, and now I have all of those noises back.

If you have manual roll down windows, you'll have to buy a special tool to remove a little clip that's behind the roller handle. This tool was included in a kit for about $7 that I bought at Harbor Frieght, but you might be able to find the kit at other auto parts stores. Google "removing window crank" and you should be able to locate a video that shows how to get it off.

Anyway, I switched the speakers and now we just don't have music coming out of the passenger door speaker, but at least we have all of the warning chimes and turn signal clicking noise again.

The dealer wanted almost $200 for a new speaker and to replace it (labor). Rediculous.

sounds like the alternator is dead and the battery is discharged now

The door sensor needs adjusting. It may be loose.
Loosen the lock nut on the pushbutton sensor in the door frame.
Unscrew the sensor a little and re-tighten the lock nut.