Not necessarily, because it's work under the table (i.e you're not getting paid) meaning you're doing it for a friend/gift type of way
"under the table" work doesn't mean you're not getting paid, it means your pay isn't reported to the relevant tax authorities. That doesn't make it less illegal.
No, if you're being compensated, that's pay. Otherwise it smacks of escorts' "You're paying me for time, anything that happens between two consenting adults ..." wink wink nudge nudge
Working for free is still working. It usually requires the right kind of visa and a work permit to remain legal. Do it on a tourist visa at your own risk.
Yes, this is an effective way to get yourself deported
And many places it's a hellish experience because before they kick you out of the country you get to spend some time in an immigration holding center which tend to be rather horrifying accommodations. Unlucky enough to get nabbed before a long holiday weekend? 3 or 4 days packed in with all sorts of people awaiting deportation, confined to your one square meter of space, with a hole in the floor for a toilet, and food unfit for human consumption. All you get from it is a story for a blog post.