I was approved for Tps on Sept.1 and now got the EAD card. Can my boss file for my green card through employment basis.

Yes, if your employer agree to do so.

H1 is not necessary to file for GC. Even though F1 is non immigrant intent, one can easily file GC. Plenty of people have done GC while being on F1.

I was F1 Out of status student for more than 3 years . will this be a problem ?

I am also in the similar situation. I think ... since you are out of status for too long ( 3 years ). In my case I am out of status since 1 month. I stopped going to school after they announced TPS. It will be tough to get GC through employment because TPS does not nullify your previous immigration offense.
By the way if you already have been approved for TPS, from what date it was effective ?

@ShankarHuShankar
I am also thinking about the same thing. I was sick of going to CPT college so I stopped going to school and was not selected in H1 in 2014, 2015. We can file GC while we are on TPS but if we have unauthorized employment history in the past, it can be an issue. It gets reset if we go back to Nepal and return on a visa (245 k). So I think TPS-> H1->GC is the better route.

wish TPS would be cancelled soon

Why ? Kina maobadi le timro bau ko chaak ma hanya thiyo ? Tesari GC nikalya ta hola ni haina timile ?

english please I don't understand what you saying dude

Oh teso bhaye gora le timro amaa ko chaak ma hanya raicha. My bad sorry.

did u say yours got knocked up by a whiteboi like me, i feel sorry about your mom

Kina kaale le timro chaak ma hanyo ? La badhai cha.