In my understanding, if you graduated and used TPS mean you are no longer in F1 . Employer can apply for H1b, but your status change will not occur. So, you have to leave the country for H1b status change...happens in H1b visa stamping. GC is possible, I guess, but I am not sure how much legal hassle employer has to take due to TPS, and this could be discouraging factor for them. I have seen Myvisajobs.com listing successful cases from TPS to GC.

Please add what info. you have or will have in the future. Good luck!

Sevis 3 months ma expire huncha kyara..can I go for H1 (Yes or No) Can I or can't I? If I can, is there a chance I can make it before deadline in April? {{Why am I asking this, I don't know, AP apply garna socheko thiye, is this a good way, thaha chaina}}

>>>> Your current status is TPS since you have already started using EAD . Once Sevis is over after 3 months, you no longer maintain F1 as backup to retain after TPS is expired (assuming TPS didnot extend after Dec 2016).
H1B: You can request your employer or sponsor company to file H1B (Bachelor or Master quota depending upon your degree) on April 1st. You still have 1 month but time is running out and you need lots of documentation and wage determination process. Go through H1B application process in USCIS and your company need to hire the Immigration lawyer to file the H1B for you. Remember, H1B approval is purely lottery base and its result will be known only in Sept.
Since your H1B is lottery base, but you may get to know on Sept, that means before TPS expires. Sounds good and if you won the lottery, you are safe for 3+3+1, i.e. total of 7 years of legal valid status to work and remain in US. (1. Check what happens when in a bench period, when you are looking for a job if previous job is terminated. 2. Check how to maintain H1B status)

Yadi company is interested (supposedly) H1 nabhaye Green Card ko route jana milcha? (hopefully) milcha bhane how do I go about this? before I start looking to talking to lawyers? Links? Forums?

>>>>
GreenCard: You can file for GC (under EB2 or EB3 ) directly from TPS but Green Card process may take much longer time than H1B and if TPS expires, you may be in trouble to maintain the status. Think about it.

My 2 cents

-- Keep your F1 along with TPS by getting into OPT.
-- With OPT you can file for H1b and GC without any hassle. For GC since you are a grad student you should qualify for it.
-- If company agrees they can file for H1b and GC at the same time. Remember GC is a long process about 1.5 years with all the steps involved.
-- For H1b you should be able to make it on April 1st which is the start date as well as the deadline(pun). If they decide to go for it ..there is not much time left, start the process now so you get the LCA before April 1st.