Am I eligible for a H1-B transfer to a new employer ?

You can only transfer your H-1B status if you are IN H-1B status and you must be in the U.S. to be in status.

The green card that i will give my wife its Conditional two years or Permanent because i will get marriage in a USA .

If you have been married for less than two years at the time the GC is approved, it will be conditional.

Traveling doesn't affect an EAD application in this case so you should be fine.

COS to F1 and now on OPT. Travelling outside US

If you travel outside of the U.S. you will need an F-1 visa

Is it possible for another employer to obtain my I797 with petition number and eventually file a transfer request

Attorney Russell is correct, you should be able to obtain a copy.

A work visa with a I-130 already approved?

By work visa do you mean H-1B? She can apply for an H-1B if she has an employer who will sponsor her since that visa does not require nonimmigrant...

What do I need for internal checkpoints in Arizona with a i-485 pending?

Bring the documents you listed and you should be perfectly fine.

Will my mom be considered a public charge because of her health?

From your statements there don't seem to be any issues with overcoming public charge.

In form i-765, question 11, which date(s) are expected to be written if one answers 'Yes, I have applied for OPT in the past'?

Provide a copy of your previous EAD to show the dates.

Will the previous case be an issue with my opt application? I have f1 stamped and opt i20

Re-submit the withdrawal the request and it should be ok.