You may retain your LPR status on reentry from an extensive stay abroad.

With your application receipt number make a case status inquiry on USCIS website.

This is a classic domestic travel. Immigration status of passenger is irrelevant. A possession of any photo identification is all you need. Enjoy!

Travels with a pending asylum case is discouraged.

Under your facts, supported by documentation a tourist visa is viable.

You are encouraged to write to NVC or make ease inquiry.

Make a FOIA request for your lost approval notice.

A properly issued B1/B2 visa is good for the entire duration of its validity.

Edibility will be based on case-by-case basis. This is indirectly an immigration issue and directly an issue for agencies that provide medical...

Very difficult to predict. It depends on local USCIS office workload, staffing, and number of cases before your I601.