I have been playing for a year and a half off and on, and I can answer some of these based on what I've been doing. Others I do not know the answers to!
What is the recommended percentage of time I should be spending learning tactics/joseki/Tsumego/master games/ playing?
The answer depends on how many games you've played. When you first start, the recommendation is to lose 50 games as quickly as possible. I enjoyed playing games on the 13x13 board at first. It's large enough to have some opening strategy but small enough that if you get disastered it won't be too long before the next game starts. At the very beginning you get more out of just playing games than any other sort of studying. You will probably not understand much about pro games.
If you've played a lot of games already, then you should keep playing and also spend a good amount of time doing problems. I just had a thread a few days ago asking about studying joseki, and a lot of strong players told me that it was still too soon for me to begin worrying about joseki. That is most likely the case for you as well.
I'm assuming that there's still opening, middle/endgame. What are the general strategies of each stage?
Here is a video that was very helpful for me on opening strategy.
Are there any shortcuts to mental calculation in Go? In chess we have shortcuts where we count the number of pieces hitting a square or can see how many moves it takes for a knight to move to a certain square.
Near the very beginning of "Lessons in the Fundamentals of Go," Kageyama famously warns the reader not to take shortcuts like this; instead, he encourages reading everything out stone by stone to strengthen reading ability. That said, we do learn what sorts of corner enclosures can and cannot be easily invaded, we learn to recognize living and dead shapes on sight, and we count liberties all the time to calculate the outcome of capturing races. I am sure there are many other things you might consider shortcuts that I'm not thinking of just now.
Are there any books you recommend? I highly recommend Janice Kim's "Learn to Play Go" series (vol. 2, 4, & 5 were my favorites. Vol. 1 was pretty basic, and I didn't get that much that was directly useful in my games out of vol. 3).
"Graded Go Problems for Beginners" volumes 1-2 kept me busy for many months when I first started. I'm still not finished with volume 3 because I keep starting over when the problems begin to take too long.
Good luck!