On top of Nick's comment, I'll add this. "Finished" is functioning as an adjective here (as Nick says). The structure is analogous to this one:
I came by to see whether you are happy with the CD that I loaned you.
Where happy is an adjective. When you want to use a verb as an adjective, you use the past participle. Further examples:
I came by to see whether you are done [satisfied, irritated, saddened, etc.] with the CD that I loaned you.
In your next picture, "passengers waiting for Flight 774" are passengers who are waiting for Flight 774. The who are is implied and understood. Also, will be departing is an example of the future progressive tense, which you can read about here.
Future progressive is used for an activity that is ongoing at some point in the future. Compare these:
I will be studying all evening.
I will be walking for most of the afternoon.
At midnight, I will be sleeping.
with these examples in the future tense:
I will begin studying after dinner.
I will start walking by 2 p.m.
I will go to sleep at 11 p.m.
Your particular example could read your flight will depart at 11:30, too. But use of future progressive suggests that the departure will be ongoing at 11:30, which implies that the process of departing isn't an immediate one.
Whether departure implies a single point in time depends on how you look at it: if the plane is considered to "depart" at the time they shut the doors, that would suggest use of the future tense. If the departure consists of several steps, such as getting the passengers on board, taxiing down the runway, getting clearance, and taking off, that suggests future progressive. Since both are essentially true, you can use either tense.