minecraft-java-edition
The Halloween update for Minecraft added an actual function for fishing rods; you can actually catch fish with them now! To do so, you right click and you cast the line.
Can fish be caught in all water tiles, or must they be a certain depth, certain number of tiles together, etc?
Actually, as Johonn and Finn Rayment suggested in the comments above, lava does not interact properly with the fishing rod. It does not burn the lure, and it just (appears) to sink to the bottom of the lava pool, unlike when fishing in water, it will float.
It does not cause any damage to the rod at all, while casting onto dry land, or any other mob or entity will cause damage to the rod.
This was tested in 1.8.1, in both Survival and Creative, (there was no difference between the gamemodes) though I am unaware of the behaviour in earlier versions. It does seem bugged however, due to the odd behaviours I described above.
To teleport a player to a movable location, you'll need to use an entity. Summon a custom named entity, such as a WitherSkull, like this:
/summon WitherSkull ~ ~ ~ {CustomName:"HomeMarker",direction:[0.0,0.0,0.0]}
You haven't detailed how your house moves, but when it does, move this WitherSkull too.
Then, whenever you need to teleport back home, run something like this:
/tp @p @e[name=HomeMarker,type=WitherSkull]
Keep in mind that you cannot teleport to an unloaded entity, although if this house is "constantly" moving then you must already have it loaded.
Best Answer
Fish can be caught in any body of water. Unlike pigs, cows, or chickens, fish don't exist until they're caught, so it doesn't matter where your water is.
You do need a minimum size body of water so that the lure doesn't get stuck on a block -- as long as the bobber is free-floating, you should be good. I've actually managed to cast into water falling from a "spring" block, have the bobber travel up to the top of the falling water, and then catch a fish.