Hello, RekhaT
To answer your questions:
1) It probably depends on what state you're in, as DMV rules differ from state to state. I live in the state of Iowa, and am here on an H4 visa. I was able to take the written test for an instruction permit. I then took the driver's test and have an Iowa driver's licence. This was done with no hassle at all (I produced my passport and I-94 as proof of identity and legal status here). It may be as easy as this for you in your state. However, your licence may only be valid for as long as your visa is. This means you will have to renew your driver's licence every time your visa expires and is renewed.
2) It is difficult, but not impossible, to get a car financed here with no credit history. The dealer from whom you're buying the car will try to arrange the finance for you. Since you have no credit history, you will probably only be able to get a highly unfavourable (to you) interest rate on your auto loan. However, you can re-finance the car at a later stage, when you have established a credit history. Community credit unions are usually more amenable than regular banks in granting loans to people with no credit history.
To establish a good credit history, apply for one or two credit cards and use them as often as possible. Pay off the full balance each month, and don't ever pay late. Also, don't miss payments or pay late on any other account you may have (e.g. an auto loan). After about a year of responsible credit activity, you can ask for higher limits on your credit card, and look at re-financing your car.
I hope this helps. Good luck!
Ansuya