My 1999 dodge Intrepid makes a loud whining noise (like a power steering pump) any time it is running... Any ideas on what it could be?

When does this occur? If when going over bumps, probably a worn out and dry control arm or stabilizer bushing; if all the time while the engine is running, probably the drive belt on the engine.

If you are sure the transmission is the source of the whine then it would be caused by a part called the planetary gear pack inside the transmission. I would have this noise diagnosed by a transmission repair shop technician, most will do the preliminary diagnosis for free before you go and put a transmission in it. Most whining noises are not transmission related but are caused by a defective power steering hydraulic pump. Bottom line be sure before you replace any parts.

Add power steering fluid

replace the power steering pump and check for any other power steering leaks (bad pump , low steering fluid )

Check the fluid level in your power steering pump reservoir. Sounds like you may have air in the pump.

just let it idle and fill till its full. it may take a while to bleed air out. turn it off and let it set a while, when the fluid is clear, adjust the level till its at the full mark.

What size engine? 4, v6, v????8 Engine code (8th VIN digit) When asking engine questions, it may help to let us know what engine you have.

Sounds like a combination of a power steering pump (whine) AND the CV axle (loud clunk noise). Check the fluid level in the power steering pump. The pump is located on the passenger's side towards the firewall. Kind of hard to get at but unscrew the cover and there is a small/short dipstick there. See what it reads. Add Power Steering fluid (NOT transmission fluid, as needed. Don't overfill)
I'd get the "thumping noise" taken care of IMMEDIATELY. Could be a serious problem. Qualified front end shop should be able to diagnose and give you an estimate of cost. Last time I had one done it was about $300 parts and labor...

use a garden hose to locate the noise hold it to your ear and touch the ps pump , and other engine parts ,but if not the ps pump i would check the altenator bearing

check alternator bearings for noise