And a follow up question:

In that first pic, you can kind of see that I have a funnel and smaller pipe wrapped around where the main pipe connects to the air handler. There are 3 holes there; 2 of them have threads and a plastic cap inserted in them, and the third is just a rectangular hole that I assume is for air and can't be plugged.

I had to redneck-engineer a funnel there so that when water pours out, it hits the funnel and then makes it to a hole directly beneath.

So the question is: why did they pipe the water all the way down and back up to the right side of the tank, instead of straight down to the hole that's like 4" away?

Cut the pipe on the left vertical piece and glue in a union fitting so it can be taken apart easily. No reason at all you would need to rebuild the whole thing.
I don’t see the funnel you mentioned. Holes should be plugged so air doesn’t come out. I don’t understand why one would be open.
They dropped the drain pipe below the pump to create the water trap to keep cold air from blowing into the pump and then out into the room. A/C condensate drains need to have a trap of some sort.

@BobH, thanks for the advice!

Here are some pics of that hole I was talking about. You have the 2 threaded holes that are plugged, but there's a third hole between the white pipe and the bottom plug that has no way of plugging. That's the one that forced me to set up a funnel underneath.

Is there a reason why they didn't pipe those 2 holes that are plugged in to the white pipe?

Good question. I haven’t seen 3 outlet holes before like what you have. Remove the lower blue cap and see if water comes out. It shouldn’t be coming out that odd shape area unless it is leaking out a crack. The water drips off the evaporator and collects in a plastic tray and comes out where the white pipe is connected. Sometimes the pipe will get clogged with black goop which is bacteria that forms in the pool of water. I like to see an “EZ Trap” screwed into where your white pipe is because it’s easy to see if there is a clog and it’s easy to clean and you don’t need a trap down at the pump- the pipe runs down right into the top of the pump.
If there is a blockage water can back up and go places where it shouldn’t and make a mess.
Hope this helps.

The reason the person who installed the pipe to wrap around the condensate pump was to create a P-trap. It looks ugly but it works. The problem is everything is glued together and if a clog develops you won't know about it until it is too late and you have to cut the pipe.

I second the recommendation of adding a clear EZ Trap. You will see the clog develop and it is easy to open and clean. On some models a clean out brush is supplied.

Usually coil cases have two threaded connections. Yours has a third unthreaded connection so a PVC can be glued into place. The drain should be at the lowest point. The second connection allows you to install a condensate alarm. This is wired so that if the drain is clogged the water level rises to the second connection and the alarm shuts down the AC. This prevents a flood and forces you to clean the drain. EZ Trap also makes this condensate alarm.