Super late, but I figured I'd answer for the archives.
Combos stop all incoming attacks from your opponent. After the time window for continuing your combo has passed the board will start dropping until either all of the pending attacks are shown on your screen or you start another combo to delay the incoming attacks. You're mistaken in saying that attacks are coming in during your combo. The odd part of this is that you may have multiple incoming attacks all queued up waiting to hit you because you delayed their appearance with a combo of your own. Unlike bust-a-move, there's no visible indicator for incoming attacks, but if you listen to the sound effects and glance at your opponent's combo counter occasionally, you can minimize the surprise.
Also, if a ball is thrown back up to the board mid-combo, the combo continuation time is extended until that ball comes to rest. This can be exploited as in this example
dkdk
xxxx
c
If "c" is the clown and "x" represents balls of the same color, the clown could extend an ongoing combo by stepping to the left, picking up "x" then going right to drop "x". While "x" is dropping the clown can move to the right, pick up the other "x" then move left to drop it while the first "x" is still falling. If done with sufficient speed the resulting matching columns of "x" will extend the combo despite taking more time than normal since this phase of the combo was extended by the time it took for all balls to settle.
Sadly, clearing the board doesn't do anything special in the vs mode of this game. Money Puzzle Exchanger, which is a similar game, gives you a big point bonus and attack bonus for clearing the board.
You also asked if the computer was likely doing big combos and the answer is they probably were and were likely also getting lots of the pieces that make all bubbles of a certain color disappear. The AI isn't very good, but they make up for it with cheating
. Keep in mind, that what character you select acts as a bit of a handicap. For instance, I play the fool and he only gets one color clearing gem and his drop pattern is mostly rows of the same color, making his attacks very easy for the opponent to clear. The Empress, who is unlockable via code, has 2 diamond color clear gems which are a wild color clear gems...also her patterns have ice blocks in them. So, depending on your character you might have a much harder or easier time beating the game at a given difficulty level.
If you want to get a feel for how much damage a particular combo is giving, you can just keep an eye on the lower right corner of the board and look for the number next to "lines". Unfortunately, the line counter tops out at 7, despite attacks being able to exceed that number.