I can't answer all your questions, but I can assure you that you probably don't have a suffiently large enough power service connection at your house to blast out a powerful enough signal to do much damage.

If you did use all available power at your address, it would take almost no time to locate the person blasting out a massive "it's me!" signal over the radio waves. Much like if you used all the power at your address to blast out an incredibly loud sonar signal, people will notice pretty quickly. Also, the massive antenna or radar dish/dome (much like the massive speakers) would give you away visually.

Someone from the FCC is going to show up pdq. It's really easy to triangulate signals. (This is the reason it sucks to be a radarman in modern warfare. whenever you're transmitting you're basically transmitting a huge "HERE I AM" sign to everyone with an antenna)

https://en.wikipedia.org/wiki/Direction_finding

You're never going to flood those frequencies except locally. Radio needs 4 times the power to go double the distance. It gets really crazy how much power you'd need not to mention the size of the antennas you'd have to have.

Edit: I guess i missed half your post.

First, is what this person claimed even a real thing?

Depends what type of drone and what type of radar but it's plausible. Some radar and radio controllers operate on the same frequencies.

Would naval radar take small, hobbiest-type drones out of the air?

Again, it's possible it just depends on some factors that we don't know.

If someone just set up an antenna in a suburban neighborhood and blasted out radar with the power of a "typical" US Navy ship or station, what effects would it have on the neighborhood?

It depends which band they use for the radar signal. Some bands interfere with local things like wifi others not so much.

Would it damage any other devices, such as TVs or computers?

Not likely anything permanent. Especially with how little power you'd be able to muster.

What effects would it have on nearby humans? Would people in the house be burned?

No effects on the humans since it's not ionizing radiation. No one would be burned unless they touched the antenna while it was transmitting. That could lead to an RF burn.

How much power would be required?

More than you're going to be getting from the house mains if you want to go past a few miles.

How long would it take the FCC to determine exactly which house had done it? How would they pinpoint the location?

See my answer above.

So just going off of what I am somewhat familiar with - there's a thing with certain wifi frequencies where at times that band needs to be used for military radar, satellite, and other radar. This seems to be sometimes a result of opening up frequencies that were already being used, with the stipulation that the new devices should stop using the frequency and give way to the original use.

The way it works in wifi, is that any device using DFS channels has some form of radar detection, so it automatically changes wifi channels as soon as it detects it. It sounds like in order to make sure they comply as perfectly as possible, the radar detection leans towards being overly sensitive.

(I double checked my facts with this technical write-up on dfs and radar if you want further reading: https://www.cwnp.com/dynamic-frequency-selection/ )

So I would wager that it's not necessary the power of the radar taking out the drones, but the drones detecting the presence of radar and choosing to shut down based on regulatory requirements.

You could probably transmit a very weak radar signal and trigger / interfere with a lot of devices due to their sensitivity - but yeah that would be super illegal, and plenty of hobbyist ham radio operators with radio direction finding equipment would be eager to track you down. I used to live in an area close to radar so while my home wifi only had to switch channels once a month, making it very worth it to be the only person on those channels 99% of the time, my workplace had DFS events daily so they had to avoid those channels.

If it’s WiFi over 2.4ghz then it’s probably a DFS channel. When the AP detects a radar signal in use it must switch channels immediately.