Daylight Savings does not cause ambiguity in flight schedules, even though it may appear to.
Basic Concepts
Before discussing your specific scenarios, let's look at some concepts.
First of all, not all countries observe DST. The ones which do, may not observe it for the same period either. Hence, airlines know if and when the destination region1 observes DST.
A flight's duration does not change because of DST. If an airplane takes 5 hours to reach its destination, it won't take 4 or 6 hours because DST is turned on or off.
Airlines and related entities (e.g. ATC) use times UTC, especially when an airplane is crossing a timezone, to reduce confusion. Pilots and ATC personal are used to follow UTC times without any difficulty. UTC does not observe DST.
When a timezone observes DST, they change the timezone code too. For example, instead of EST (Eastern Standard Time), they'll use EDT (Eastern Daylight Time).
When DST ends, clocks are moved from 3AM to 2AM, not from 2AM to 1AM.

Your Questions
Now, let us talk about the scenarios you mentioned.
If a flight is departing at 2:30AM (time corrected as I mentioned above), it will depart most probably on the first 2:30AM. Flights have their schedules mentioned in UTC as well, so it will not a confusion. As I do not know everything, I have never seen a flight departure time in that particular hour. Would you care to share an example?
If a flight is arriving at supposedly 2:30AM on the day when DST is starting, it will not arrive at 2:30AM but at 3:30AM. Because, as you said, 2:30AM does not exist on that day for that timezone.
Examples
As suggested by fooot, I'm adding an example. This flight is from JFK to AUH.
This first example is on November 1st (when DST is in effect).

Departure Arrival
EDT 12:40 EDT 01:10
UTC 16:40 UTC 05:10
GST 20:40 GST 09:10 (GST: Gulf Standard Time)
Flight Duration: 12:30
This is the same flight on November 2nd (when DST is not in effect).

Departure Arrival
ET 11:40 ET 00:10
UTC 16:40 UTC 05:10
GST 20:40 GST 09:10 (GST: Gulf Standard Time)
Flight Duration: 12:30
Footnotes
1: Not all parts of a single country may observe DST. In USA (for example), most parts of Arizona do not observe DST. I agree with them, DST does not make any sense or provide any benefit in today's world.