In the most recent T20I between England and New Zealand, only two balls were bowled before the match was called off.
However, in that time, Lumb scored two runs, and then was caught behind.
Does this still count in his career stats officially, or do all results get ignored? Similarly, if you scored an unbelievable 250 (ie it'd be the highest ever score in an ODI/T20), and then the match was called off, would it get ignored, officially?
Yes, it is still counted for his career stats. Just because a match is called off doesn't mean that the stats wont be recorded for that match. It will be still recorded as "abandoned due to rain" and any runs scored or wicket taken will be added to the stats of corresponding players.
The proof can be seen in this stat page of Lumb. You can see that in the innings by innings list, his score of 2 is also added and the total scored by him in that series is shown here as 31(29 + 2) at an average of 15.50. If it was not included, his average would have been 29 from the first match.
The same applies for your second scenario as well. If a batsman scores 250, he will get the credit for it and it will be added in his career stats.
