I also like the idea of the "back entrance"... You will have some very surprised sheep!
OK, here is how to do your calculations, roughly.
Rechargeable AAs are around 2300mAh. That means, they will provide 2300mA for 1 hour, 230mA for 10 hours, 23mA for 100 hours, and so on. (Actually, it's not that simple, but it's good enough to get you started!)
So. Assuming a 4.5V power supply, that is 3 AA cells in series (4 if you need 6V for a 5V arduino). Either way, that will give you about 150mA for continuous operations over 15 hours. If you double the number of cells (3 in series x 2 sets) you'll get 300mA to work with over the same time, and so on.
A typical GPS shield will draw around 60mA continuous (plus a bit extra for startup, resyncs, and so on). But it depends on the shield and the GPS chip.
So using that example, you have about 90mA to budget for everything else, if you only use a single "string" of cells, or 240mA to budget if you use 2 strings and so on (I hope this is making sense!)
From the specifications, a Uno board will draw (very roughly) 50mA continuous @ 3.3V. I don't know enough about the different boards to know if that's a maximum, or an idle, or what - perhaps someone with lots of knowledge can help us understand the current draws for the different modules in various modes a bit better?
But from my knowledge of other MCUs, it's reasonable to assume that the MCU will draw very roughly 60-120mA if it's running code all the time, using timers, interrupts, and so on. In your case, where all you need is to log data every few minutes, and sleep between, I think you could say much less - in that case, I would average 50mA as a starting budget.
There is a very easy way to get all the numbers, once your code is working. Connect up all the bits, your temp sensor (don't forget your "back passage" sensor!), your GPS shield, and so on, and measure the current drawn. It's as easy as that. That will tell you exactly how much the whole system is using, and from there you can estimate whether you need 3, 6, or 9 AA cells.
So, as a first approximation, let's say the following :
1) GPS Shield : 60mA
2) Uno module : 50mA
3) Flash card read/write : 10mA
4) Temp sensor : 5mA x 2 = 10mA
5) WIFI shield : 50mA, plus 150mA for 10 secs every 10 minutes = average of, say, 75mA
Add it all up : 205mA (roughly) per hour.
So, using a set of 3 or 4 AAs, you will get about 2300mAh/205mA = 11 hours and a bit.
If I were you, I'd use 6 or 8 AAs (assuming the sheep don't mind), that way you will be able to leave it going all day, and maybe all night (if it's cooler, it will use less power).
Does this help? Can anyone think of better figures to use for the different modules/shields?