120 mm fork with a frame designed for 100 mm fork? I have a Cube Reaction Team 2011. It came with a Fox Alps 32 fork, 100 mm travel (26"). The fork is now ruined and I need a replacement (on a rather tight budget) and I found a good price for a Rockshox Reba RL Solo Air fork. It's 120 mm, but I think it can be adjusted to 100mm - I'm wondering though, would the extra 20 mm travel be a problem for my frame?

My point is basically these two questions:: firstly, would it ride/handle worse, and secondly - would it put more stress on my frame, especially if the stress could be potentially fatal for it (it's a carbon frame and it's one that's difficult to get anymore, so I'd rather keep it intact).

Also a side-question - is there any possibility that a 100mm, 26" fork would NOT fit my bike? I probably cannot install the fork myself and I'd rather not buy one and then find out at the bike shop that it cannot be installed on my frame.
Before starting, your fork looks pretty nice. Are you sure you can't send it to FOX and rebuild it? Why do you say it is ruined?
About the fork size, please check the cube FAQ:

CAN I INSTALL FORKS WITH MORE TRAVEL AS PROVIDED IN THE STANDARD EQUIPMENT?
NO. The installation of a fork with more travel will void the warranty. The bike geometry will be changed in a negative way, which leads to a different handling of the bike and can damage the frame permanently.
It is also not approved to install double bridge forks into CUBE bikes.

If your bike is still under warranty, I don't think it is worth it. If it isn't, it will for sure put more stress on the frame but I don't think it will brake. This depends a lot of your weight and ride style, and keep in mind Cube says not to do it, so it is your entire responsibility. I wouldn't do it.
I am pretty sure you can change your suspension between 80, 100 and 120mm of travel. Please check here. Rock Shox has really good manuals so you might even be able to do it yourself, otherwise take it to your bike shop.
Changing a suspension is simple, if you have the patience and the time to learn, and you just need one or two allen tools for most of it. However, sometimes taking the bottom part of the headset from the old fork is a pain. Search on youtube for fork replacement. If you feel like it, I think you can try and if something goes wrong take it to a shop in pieces. They will figure it out :)
Before buying, confirm with the seller the fork will fit the frame, specially if it is an used fork he might have shorten the part of it that goes into the headset and it's shorter size might not fit your frame or your favorite stem position.
IMPORTANT: Please see @mikes comment below.