My opponent has just captured my last town with his general and my capital with a cav unit in the same turn. Should this be possible?
Firstly, no your opponent is not allowed to capture the last town and capital on the same turn.
5. The Cards.
...the player keeps the card until the town is captured by another player, when it must be surrendered.
6. Capturing Towns
Only a General can capture a town - simply by moving on to the town square. When a town is captured, at the end of the turn the player claims the town card...
7. Capturing a Capital
A Capital can only be taken after all the provincial towns in that country have been captured and are still held...
Rules 5 shows that a player must hand over a card once a town is captured and rule 6 states the card is handed over at the end of the turn. Together these show that possession of a town is only handed over at the end of the turn.
Secondly, I do not believe that any piece apart from the general is allowed to capture the capital:
Rule 6 (quoted above) says only a General can capture a town.
Rule 7 (also above) says a capital can only be taken after all provincial towns are captured.
What are provincial towns? Rule 6 only refers to towns. I believe, the word provincial is introduced in rule 7 to make it clear the 4 'non capital settlements' must be captured before the 'capital settlement' and therefore both types of settlement are included under the term 'towns' in rule 6.
You or your friend may wish to interpret the rules differently but I believe the intent is clear. Towns and Capitals may only be captured by a general and only one per turn.
Hope this helps,