How to send additional proofs of marriage for H4 stamping (221g | Yellow slip)?

Inquire with the consulate in question directly.

Will be moving to a different state within 15 days after sending i751 application.

To avoid unbelievable complications, delay and frustration wait to first move, then file the I-751.

Either date will no - will not make no difference.

Currently, I am on my STEAM OPT,

Pose these questions to your H-1B lawyer - he/she is paid for precisely that.

Can sell your own stuff, however cannot run an Etsy business in the USA while on a work visa.

Filing for husband to immigrate to US

There is no way to properly advise you on such a sensible matter on a public blog that Avvo is.

I got married on a J1 visa and this one expires in October. If I file my adjustment of status now can I still work on this visa?

Yes, you can still work on J-1 visa as an au pair, if that's what you still want.

As an asylee who applied for green card you will not be asked to appear at any AOS interview - you will receive your GC in the mail.

Can I file for my mother who came in to the US legally but overstayed?

yes, absolutely do so at once. You may need a co-sponsor for the I-864.

Am I already a US citizen as someone suggested? If so, what do I need to do to get that evidence?

I agree. Apply for a US passport and go from there.