Travelling a month after applying for Green card through US citizen

You should wait till you get the advance parole through your 485 filing else the 485 will be considered abandoned.

No, that is considered work. Working on a H4 is not permitted.

If you applied for an extension, your previous visa might still be valid. If it is, you can use it to travel. Did you file for a travel document...

The status showed Initial Review and Now it is showing acceptance

It means the file was transferred to the pp unit.

How can I help my boyfriend

If he was deported twice, there is not much you can do. I would recommend hiring a local attorney to review his case with you.

I received a letter from USCIS that they temporarily closed my application to re-new my work authorization card. My question is

I agree with my colleagues.

Getting married to my fiance who's a citizen while I have F1 Visa

You need to file for your 130 and adjustment application before your F1 expires. The employment based green card will not affect your family based...

I got confused what to write on the form I-485 because it asking about my current USCIS status and the expiration of it.

If K-1 is your current visa, then yes, the expiration date on the I-94 that was given to you when you entered on the K1.

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I-94 expiring before I-797 H1b approval due to passport expiry

You need to file for a H1 amendment or a I-94 extension. Your employer's attorney should do the filing.

You can request that the name be corrected.