Yes, you can use any website to file your taxes. Your each year W-2 printed forms are proof of your yearly earnings.

I am not a resident. Can I still use Turbotax?

After 5 yrs, all are resident for tax purpose.

As far as I know, to be a resident for tax purposes one has to pass Substantial Presence Test. Some visa categories are exempt from this. For example, F1. But, if F1 student is here in US for more than 5 tax years, he/she can take substantial presence test, and if he/she passes, he/she will be considered a resident for tax purpose.
If you were here in US from 2009, you can't take substantial presence test till 2013. You can take substantial presence test in 2014, and to pass it, you need to be present in US for at least 183 days in 2014. Since you have to file tax by April 15, you will not satisfy this condition, and hence can not file tax return as a resident for 2013. However, if you stay in US for more than 183 days in 2014, you can file tax return for 2014 as resident (which you will be filing in 2015).

This is what I understand, but I might be wrong. Please consult a lawyer or call IRS to get the correct information.

You can find more info here:
http://www.irs.gov/Individuals/International-Taxpayers/Substantial-Presence-Test
https://international.uoregon.edu/isss/taxes/info/SPT

I have no idea how many years tax return are checked during Green card process.

TTT

Student (F-1) can file tax using turbo-tax or using any other software products. The only difference is non resident cannot claim Earned Income Credits or any Education credits. If you filing only your W-2 return, you don't need to dig more information.