You do not need to ask USCIS to correct their records. If you got the green card, you have what you need.

Since the H1b category is a dual intent visa, it is not an issue to be married to a US citizen. They will not deny it for that reason.

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Different lawyers charge different fees. The employer will have to pay so they should find the lawyer also.

If the purpose is to get a faster decision on the H4, no, it won't work.

No. The grace period gives you time to leave the U.S. But if you are not applying to enter the U.S. as a G-4, you cannot be admitted as such.

No. The earliest you can file is in March or April next year.

Since this category of visa allows "dual intent", meaning, she does not need to prove she has no intent to immigrate to the U.S. if she is...

That is what it means. Whether that actually happens may or may not be true.

No. But he could try to get a B1/B2 as a domestic partner.

Yes, so long as the visa bulletin shows that category F2A is current and the foreign national is in lawful status.