Changing cassettes for hills

In short no. You'd have to give us some more detailed specs for better numbers (e.g. make and model of your derailleur) but typically a short cage derailleur would have a max capacity of not much more than 30t. As it is, your current setup requires a capacity of (52-42)+(25-12) = 23t, so it's fine. Going to an 11-34 would increase the required capacity to (52-42)+(34-11) = 33t, which exceeds the max capacity.

In addition, derailleurs also have a maximum size of cassette they can accommodate; I can't think of any short cage derailleurs which will allow a cassette more than 30t in size. At a minimum you'll need to change your rear derailleur to a medium cage.

For example, the Shimano R7000 medium cage derailleur will accommodate a cassette up to 11-34, but you'd need to find the appropriate derailleur for the rest of your drivetrain.

In addition to the points made by DavidW regarding capacity limits of a rear derailleur--which can be exceeded with good results if the overage is not excessive. To me this means exceeding total capacity by 4-6 teeth or less and not going over max low sprocket size by more than a couple teeth. In the right system, going 4 teeth over max sprocket size can work but in most, slow, noisy shifting that gets the chain onto the too large cog 2/3 of the time would be what you're dealing with. It works but not well, so, why?

When one changes the size (commonly the unit of measurement is number of teeth) of an aspect of the drivetrain, whether front or rear, the chain will also need to be resized for best performance. It also avoids possible ruinage of the rear derailleur that can happen when a too short chain is used. On the other side a loose chain makes more noise, derails easier and more often, can cause poor shifting, and also requires the rear derailleur to take up more chain slack, which may not be possible.

Thus, your jump from 11-25 to 11-34 is excessive, probably won't play well with your short cage derailleur and would certainly require a chain resizing even if it were doable. If lower gearing is what you're seeking at a minimum of cost, look to changing your front chainrings to something like 48-32 or even 44-28. Considerations for doing this include having a crankset with removable chainrings (some cranksets feature riveted chainrings which require the whole crank to be changed). If your chainrings are found to be bolted, you must determine the number and spacing--termed the B.C.D. or bolt circle diameter--of the chainring's bolts and then use this information to see about finding compatible replacements with fewer teeth. The change will require resizing the chain--in this case, taking some links out, to avoid problems mentioned above. A further option is to buy an entirely new crankset. More cost than just chainrings (given similar quality levels of the two) but today's cassette prices what they are, you'd be 2/3 of the way to the cost of a new crankset getting the proposed cassette, which as we've discussed, has a poor chance of working anyway. So look to make some changes up front instead.

Almost certainly not. A jump from 25 to 34 almost certainly requires a long cage derailleur.

You need to look up the max teeth capacity of your derailleur, they are usually specced. You can calculate the capacity you need (or the max capacity you can go to with a current derailleur) like so: (big-front + big-rear) - (small-front + small-rear). In your case that's (52+25)-(42+12) = 23 teeth capacity. That is, if your derailleurs has a teeth capacity of, say, 26, you can go 3 teeth more and get a 11-28 cassette.

Other than that, Rule 5 :)