How to amplify 3.3v to 12v (PIR and led strip light)

The FET you link to is not suitable for your application. Taking a look at the "typical characteristics" table, we see this:

Vgs(th) is listed as between 2 and 4 volts - already it could be higher than our supply voltage, but not especially the "conditions" column on the right: the specification is given with only 250 microamps of current. Your LEDs, presumably, will draw a lot more than that.

To figure out if the FET is suitable, you need to know how much current you plan to draw, then look for the graph plotting "typical output characteristics":

Look for a line corresponding to the gate voltage you plan to use - in this case, there isn't one low enough - and look for where the curve flattens out. If that's at a current level greater than the amount of current you need to supply, you're good. Alternately, look at the "typical transfer characteristics":

This plots maximum current as a function of gate voltage, so you can trace a line from your MCU's gate voltage - 3.3 volts - to determine what the maximum current level at that gate voltage will be. Again, in this case, the plot doesn't actually go low enough to evaluate that on this particular FET.

Keep looking - there are FETs that meet your requirements - and remember that everything depends on how much current you plan to conduct.

One single large battery will be easier to deal with, and will also make it easier to switch all the lights on and off.

To determine how big a battery you need, do the math. Add up the current requirement of all the lights. That's the current the battery must deliver. Now divide the battery capacity by that current to get a time. That's the time, in theory, that the battery can deliver your current for. However, for off the shelf 12V lead-acid car batteries, you want to avoid running them very low because that will cause permanent damage. Either get a larger "car" battery, or a lead-acid battery specifically designed to be able to be drained to nearly empty. Such batteries are called deep cycle, or are sometimes sold as marine batteries.

For example, let's say all the lights together draw 1.3 A (just picking a number out of the air). A 50 Ah car battery can supposedly run the lights for (50 Ah)/(1.3 A) = 38 hours. However, let's consider only half that actually usable for a consumer level car battery, so 19 hours. You'd need to get a bunch of such batteries to last the 112 hours you want. In fact, (112 h)/(19 h) = 6, which is how many 50 Ah "car" batteries it would take.

Since you say you only need to run the lights 9:00-17:00 each day, a better solution is to use a single battery and charge it overnight. Presumably the gallery or whatever is closed during the off hours, so it doesn't matter how it looks with a cord running to a outlet during this time.