Let me clear you about TPS, F1 and Green Card Process.

1. Since you are in F1 and if your TPS is approved, and if you have EAD, you can join company if you have bachelor degree or above (transcript equivalent needed if you studied abroad, not in US.)

2. Once you join company with TPS status and use EAD, your F1 status should be on HOLD (provided you have maintained the minimum credit as advised by the International adviser and SEVIS status did not change.)

3. Once you worked for the company full time with W2 (40 hours/week), your company should be able to process Green Card for you under EB2 (Bachelor's degree) or EB3 (Master's degree ).

4. Petition for Green Card Process is a lengthy process and your company should hire experience immigration lawyer. This process might take anywhere between 1.5-2 years, provided TPS extends for another 2 terms in your case if your Green Card has not yet started.

Green Card Application is a complex process and has its own flow.
Step1-Prevailing Wage Determination (PWD)
Step-2: Advertisement
Step-3: Recruitement and Selection
Step-4: PERM (Labor Filing)
Step-5: I-140
Step-6: I-131,I-765, I-485
Step-5 and Step-6 can be filled concurrently.But there is a risk if I-140 is deined, all money you spend for Step-6 will be lost. It's a huge loss. So, if you are 100% sure, your I-140 will be approved, then only go for concurrent filing otherwise wait for I-140 approval before filing for Step-6.

I hope you got fairly good idea ? Let me know if you have any question. Some people might argue that Green Card through TPS is not possible, but again this is a grey area and as long as you are eligible for employment, green card process is a future employment and might work out with TPS too. However, I would suggest you to consult a good lawyer before jumping into the lengthy and time and money consuming process of green card for peace of mind.