tel Aviv airport lounge
Yes the private airport lounges, Traklin Dan, in concourse B and C have a Teudah from the Rabbi of the Airport. The El AL lounge in Concourse D has one from the El AL rabbi.
The are plenty of restaurants and cafes with a Teudah both pre and post security. For a more quiet atmosphere if you have the time, you could go downstairs to the arrival hall on floor G where there are also some Kosher cafes
Yes, lots! 2 years ago, I couldn't find anywhere with a teudah. Last time we were there (~8 months ago), we brought food with us from town, but it turned out that almost every airport restaurant had one. Unless something has majorly changed since then, there should be plenty for you to choose from.