In general, is it allowed for a single bowler to bowl two (or more) consecutive overs in international matches?
No, it is not allowed to bowl chain overs in any Cricket matches.
See the law 22.1
1.. Number of balls
The ball shall be bowled from each end alternately in overs of 6 balls.
Is it not allowed from the beginning of international cricket history or did the ICC banned it recently?
It is not allowed from the beginning of Cricket history.
Does the first bowler allowed to bowl the remaining balls of this second over?
No a bowler can’t bowl the remaining second over because as per rule he can’t bowl two continues overs.
See the law 22.8
Bowler incapacitated or suspended during an over
If for any reason a bowler is incapacitated while running up to deliver the first ball of an over, or is incapacitated or suspended during an over, the umpire shall call and signal Dead ball. Another bowler shall complete the over from the same end, provided that he does not bowl two overs consecutively, nor bowl parts of each of two consecutive overs, in that innings.