Two fields in DS-2019. One is in the skill list , another not.

Depends on the program. Speak with your sponsor or take to an immigration attorney for review.

Can I accept X company offer

1. if meeting the requirements for filing are met as noted by my colleague 2.No 3. Yes

How much is an average lawyer's fee for this type of service & how to choose the best one?

Each attorney may set their own fee and there is no average fee.

Explanation about USCIS online case update "my Case Approval Was Certified By USCIS"

The update is unusual. You may wait a few days for a written decision and if none forthcoming consider an infopass appointment to seek clarification

If you filed for employment authorization and advance parole the EAD and AP document should arrive next.. The interview several months later.

Generally no consequences for housing a person whose visa expired unless it rises to a level of a criminal act under 8 U.S. Code § 1324 - Bringing...

Green Card Holders Having a Baby outside US

8 CFR § 211.1 Visas. …(b) Waivers. (1) A waiver of the visa required in paragraph (a) of this section shall be granted without fee or application...

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Can a parent of a US born child living in Canada obtain custody of that child.

You should contact a domestic relations attorney in Canada and the US for child custody issues. This is not an immigration question.

Can I start working for a profit company at April 1st, if I get a new H1B from them, and I have a cap-exempt H1B at hand?

An H-1B may port from a cap-exempt employer to a cap-subject employer during a period when there are no H-1B numbers as long as the cap-subject...

.Is there anything I can do to make it faster???

See the USCIS website for expedite criteria. If you fit the criteria you may ask for expedite.