Can I replace my existing Shimano 12-25 10-speed cassette with a SRAM 11-32 10 speed cassette with no other modifications?
If you have a 10 speed 105 group it will be either 5600, 5700 or 5701 series. Looking at Shimano's archived specs from here, 5600 appears in 2006, 5700 in 2011 I believe, 5701 is in the 2014 doc.
The RD-5600 has a max rear sprocket of 27 for both short cage SS and long cage GS versions. The RD-5700 has a max rear sprocket of 28 for both SS and GS versions. A bigger cassette would likely exceed the total capacity as well unless you have a GS which has a higher capacity for use with triple cranks.
RD-5701 SS has a max sprocket of 30, GS max sprocket of 30 with triple cranks and 32 with double, so if you happen to have 5701 GS double you'd be OK.
The answer is yes.
SRAM and Shimano cassettes share the same hub uptake pattern and sprocket spacing and are interchangeable. As are chains from both makes. The maximum teeth count for the short cage version of the 105 derailleur is 30 teeth and the minimum is 11 teeth. So, go ahead. The only caveat is that your chain could be too short and needs replacing, so check this point.
BTW: the cassette tool aka lock ring tool is also compatible.
The previous answers have already explained the detailed stuff, so I'll just get straight to the practical point.
Can you? If you use a longer RD hanger or a goat link and a longer chain, yes.
Will it work as smoothly? Probably not. (especially when shifting to high gear/small cogs)
Should you? That's up to you.