Can my friend extend her L1 visa status through her company In the US without having to leave the US?

her employer may file an I-129 L-1 petition with request for extension of stay without her leaving the US

I forgot about immigration court a couple of yrs ago now they wont renew my green card what should I do? should I call ice?

you should call an immigration attorney experienced in going before the IJ and ICE

Timeframe to receive hard copy of h1b in premimum case.

hard copies follow 7-10 days from approval by snail mail the effective date can be the date approved or the date on the petition whichever is later

Can my cousin file my green card once he become citizen.

Unfortunately, your cousin is not a qualifying relative.

effective date is usually next day 3/19/2016

How much maximum time extension do I have on my H1 B?

An H1B clock is reset by the filing of an H1B cap case. When extending the old H1B you do not reset the clock, but capture remainder time left of...

USCIS locations

it depends upon the basis of the I-485 and if it is a stand alone application. https://www.uscis.gov/i-485

When uscis will ask for the original documents? Or should I send them now?

plan to take originals to the consulate at the interview

Will my new employer be able to file H1 once my current application is approved? OR do I need to wait until October?

you must wait until after your first paycheck in October

So can i still fixs my papers with out him

I recommend you meet with an attorney. You seem to qualify for DACA. You may qualify for other categories.