Do we need to fill any forms with the sponsoring employer after getting EAD.

What do you mean?

If you are not planning to use your EAD card, you don\'t have to tell your company that you have EAD. Obtaining EAD card doesn\'t mean you have to use it.
 
Employer will need a copy of EAD if H1 limit has expired...

Employer cannot legally employ you without proper supporting documentation showing your valid employment status. You can use your EAD to moonlight without informing your primary employer if your H1 is still valid and you have not used an AP to enter the country. You can even start your own business with the EAD.
 
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