While widespread stops are not commonly reported, they are possible. It is best to avoid optional travel. If you must travel, carry your passport, I-94, approval notices, and proof of your ongoing status (like pay stubs or transcripts).

It is very difficult. Because the second petition (the "bridge") is failing, an approval within the US for a third employer is unlikely. You may need to travel abroad for a new visa stamp, but be aware that fraud allegations are serious and can lead to a permanent bar from the US if not cleared.

This is a common bottleneck. Have your employer contact their local congressman’s office to push for adjudication. If that fails, reaching out to a litigator to threaten a lawsuit against the US attorney's office can often resolve such delays.