To add to what others have said, you can even apply for employment authorization under I-765v, so you have options.

Can my wife leave the country as a B2 tourist while our marriage petition is pending? I am a US citizen.

She's going to need to get advance parole approved, or else her I-485 will be voided. She could still immigrate, but it'll be a bit complicated....

I am a US citizen. Can my wife (on an F1 visa) leave the US while our marriage petition is pending?

If she has a pending adjustment of status, it will get voided if she travels unless she has an approved advance parole.

Write the date and label them by describing the occasion. It should be fine.

H1B TRANSFER with receipt number only.

Get your case number, and go to USCIS's page, and print the proof of approval.

How to fill the form i589 application for asylum form?

You'll need to setup a consultation with an immigration attorney to go over your questions. Also, I highly advise you hire an attorney to prepare...

Do I need to maintain my student (F1) status while waiting for adjustment of status? What are my options?

It's fine to drop out, but if your I-485 is denied for some reason, then you'll be out of status.

No need to accompany your spouse, but you can go if you want for moral support.

It's likely to be inconsequential, and a minor thing that won't cause problems later. It should be viewed as a harmless error.

H4 Visa valid date?

Convert to F1 status. You can't maintain H4 past age 21. Also, you should have 1 year of status on F1 in order to qualify for OPT. So get this...