Serak, a few points. I'll try to stay away from any spoilers.
1. Do you realize that it takes 5 victories to complete each "rank"? You have to play that little game of tic-tac-toe 5 times for each rank and there are 10 ranks. To complete a rank, you have to fill in the board with 3, 4, 5, 6, and 7 X's. That's a total of 25 solo battles for each rank, 250 solo battles in total.
Not to scare you away.
It just takes a lot of patience and work to finish off the entire Battle Squares.
2. Another FYI ... you only are *required* to complete 1 rank to complete the ice Temple. Finishing all 10 ranks is optional.
3. Yes, elementals are very tough.
4. Here's the strategy I use for battlesquares.
Select your very best warrior. Give this warrior the very best stuff you have. it's a good idea to have him armed with your best slashing, piercing, and blunt weapons you have, so that you have some options against whatever foes you face.
Without knowing the stats of your characters, I'd choose the paladin, for the improved saves and a little extra self healing.
Buff up this character to the MAX. Spare no spells. Stoneskin is a must. Haste is useful as well.
Don't bother with summons. They'll just get teleported in with the rest of your party.
Start the battle. Monster knowledge is useful. Select the best weapon for dealing with the monster you face. Blunt weapons for skeletons, etc.
One mini-spoiler. IIRC, the prize for completing rank 4 or 5 is a club that is **VERY** useful for dealing with those elementals and other monsters to come.
And BTW, you do not have to complete the ranks in order.