I've been keeping it pretty well watered. Do you think it'd be happier in the ground?...

Lychee does like a lot of water but it sounds like you need to give it more sun. Seedlings might grow a slower until it gets going and potted plants grow slower than ones planted in the ground. I noticed your zone is 9...young lychee trees are very susceptible to freezing temps so if you are able to provide a winter shelter for it during temps below 32 degrees then you can plant it in the ground...if not I would keep it in a pot and place the pot ( slowly acclimate it to full sun) if its not use to being in HOT full sun all day...acclimate it slowly over a 10-14 day period extending the sun exposure by appx an hour each day or you will risk the leaves being sunburned...Good luck! Also lychees are very susceptible to fertilizer burn...so when you do fertilize I would give it a very weak dosage very 2 weeks at 1/4 strength like Fish emulsion or an all purpose...light, weak dosage is better.

So, it might need fertilizer, but just a teensy bit. Should I water it with the water from my vaguely pondlike object, or would that cause salt problems?

It seems like I have 2 basic options at this point: 1. put it in the ground now, where it is, or 2. move it to the back yard for the summer sun, let it get a little bigger, then put it in the ground next fall or at some later point. Which do you think would make it happier?

Winter frost is not really an issue. We get freezing nights maybe 2 or 3 days a year, just barely, and I have lights to help it out.

Use Rain water if you can get it or distilled water if you are worried about chlorine and salts...Lychee trees are very prone to fertilizer burn that's why 'less' fertilizer is always better...I add it about every 2 weeks but in a very weak form...

I would think planting it inground if it gets mostly morning sun is best...but again make sure its not full sun ALL day...as it can burn the tender new growths...I've used shade cloth before if necessary...

Ok. Here's my lighting situation.

Back yard, I can give it morning sun, afternoon shade, but plenty of direct sun, all summer. In the winter, it will get no sun whatsoever there.

Front yardlet, there's a head-high wall to the east a few feet off, a shed directly to the south, a house a few feet to the north, and a carport a few feet to the west. So it will get either a little direct sun, or no direct sun but some bright indirect light, from probably about 10 to 2.

Yeah, my lighting situation kind of sucks.

And our water is extremely hard, just in general. Relying on rainwater is not practical (because, well, desert), and I can't really afford to give it distilled water. Maybe temporarily/occasionally as an emergency measure, basically, but in general... no. I just wanted to avoid adding insult to injury by giving it hard water that had also been evaporating for a few weeks, which is more or less what the water from my vaguely pondlike object would be.