I accidentally closed my tab with the paragraphs I've written so far. >.> Anyways...
What is the recommended percentage of time I should be spending learning tactics/joseki/Tsumego/master games/ playing?
As you are a beginner, I'd recommend just playing many games with an occasional end of game review to get a 'feel' for the game. Tsumego simply refers to go problems, and tactics would fall under that. The two main branches of tsuemgo are life and death and tesuji (similar to chess tactics I guess) problems. Professional/master games are not worth looking at in regards to improving unless you're a very strong player. The reality is most players can't understand a lot of what it is going on in any random pro game and required detailed commentaries from a professional/top amateur player. It's fine to look over pro games for entertainment but I can say I have never really gotten much benefit from studying pro games. The best way to improve is from playing and reviewing your games with strong players and doing go problems. Joseki is something you should understand rather than memorize, there's no use in learning them now. Joseki are common positions that lead to generally even positions in certain positions. The stronger you are, the easier it is to deliberate on a joseki choice in a position (or not the play joseki at all).
I'm somewhat confused as to how Go strategy works. Is it mostly protecting your own groups while restricting the development of your opponents?
This is the right idea, but oversimplified. I think you're underestimating the amount of attacking and fighting that happens in go and how the exact placement of stones can change a position drastically. There's a lot of subtlety in go.
I'm assuming that there's still opening, middle/endgame. What are the general strategies of each stage?
I'm too lazy to re-write this part.
What are the older top players I should be looking at? I'm assuming that I won't really get modern theory and should looking at older games first.
That's the wrong assumption, older games are just as hard to understand as modern games. I've mentioned in my first response how most players required a detailed commentary to appreciate professional games. The basic reason for not studying professional games for say a 5k is simple: if I showed a game between two 5d amateurs and passed it off as pro, they'd believe me. Similarly, if I showed a game between two professionals and passed it off as a game between two 5d amateurs, they'd also believe me. The fact is most players can't distinguish between a professional's play and a top amateurs play (I could tell a 5d amateur game vs. a pro game, but I probably couldn't tell between a top amateur and a pro).
How has Go strategy changed over time?
The go equivalent to hypermodernism is Shin fuseki. http://senseis.xmp.net/?ShinFuseki
Are there any shortcuts to mental calculation in Go?
Not really, you just have to read it or know it by heart.
Are there any books you recommend?
Once you get to say 10k: Life and death, tesuji, Lessons in the fundamentals of go are the best English books IMO. Opening theory made easy is also pretty good. Once you get a bit stronger than 1k: Attack and defense.