I'm on L2. I worked for 3 months without an EAD card- not knowing that it's needed. Now what?

245k would apply if the I-485 is employment based. This issue will not cause a problem. Travel outside the US will likely clean up your record.

Is telecommuting possible while being on L1B ?

Telecommute may be possible. You should consult with your immigration attorney to know how.

Do multiple, back-to-back long trips (over 180 days) outside the US cause problems, if they happened over 5 years ago?

if this travel history is within the 5 years USCIS will question your intent. Although, you may still meet the physical presence requirement.

Extension of H1b status

Yes, you should be able to apply for the beyond the 6th year extension.

I am an E2 dependent

You may not legally continuing working on an expired EAD. It is very important to file the I-765 early to avoid a gap in employment authorization....

N400 ,citizenship

Your residence is LA. Rely on your parents to keep you informed.

You are subject to your original 6 yearH1B limit. The school should sponsor you for a green card asap.

Unused and Expired stamped H1B

Although, we had had some past success filing beyond the 6 year unused H1B as cap exempt change of employer. Not these recent days.

Can I start a business on H1-B visa? What are the limitations?

You may start a business, but cannot grow the business by working for it. There are ins and outs you may consider as well. You might consult an...

What happens to your green card application of you leave US for few years.

You should meet with an immigration attorney to advise you about the timing of the options available to you. I'm not certain you are banned. But,...