My dodge stratus squeels and is hard to turn steering wheel at first start

Could be as simple as loose power steering belt, or pump beginning to fail.

Depending on your description of grinding could be power steering pump if whinning noise if tight on steering check steering rods tie rods and also check cvr joints for front wheel drive also depending on the direction of turning and which way is worse check the struts might be binding on

the pump dies when the engine stalls or runs too slow.
so, the steering has nothing to do with that , unless the
pump seized, so drop the belt and start car.
if car will not start, that is a no start question
post it as, cranks fast but will not start. and what engine option
some have diesel.

Have you checked the power steering fluid? Do the belts squeel when you turn the steering wheel?
I know this might sound silly but are your air pressures in your tyres correct? Low air pressure will cause your steering to shake at speed and it will be hard to turn at low speed.
I hope this helps,
Scott.

does it make noise when turning ? when installing new power steering pump with both front tires in air on jack stands or jacks start and fully steer right fully steer left repeat a couple times and shut off now pump is bled place on ground and test....by turning take a drive....

Try turning the steering column really hard one way or the other while you are turning the key

take the key out turn steering wheel make sure its locked put key in try again happens on my stratus every once in a while

the tyres must be rubbing on the mudgards or the shocks

could be low power steering fluid or bad pump

Sounds to me that the power steering belt is loose. If it is a 2.4 litre engine it has two serpentine belts. You should replace the belts especially if it was smoking.