"Lame" isn't used very much as a verb, and when it is, it has to be transitive, with the meaning of "render lame", i.e. to actively disable the use of a limb. For example, "the car lamed the cat when it ran across the road and was hit". So I don't think that's what you are looking for in your examples. "Limp" in contrast is intransitive, and it is what someone does when their leg is stiff, injured or otherwise not good for walking. It can also be an adjective, but that isn't used in the context you are talking about: "this lettuce that has been in the fridge for the last week is limp now". You probably want the adjective "lame" in other situations: "the cat is lame from its injury".

When I was a kid I first learned the word lame to refer to things/pepole that weren't cool or boring things/people. Then I found out it was used to describe animals that didn't function well. Like a lame horse.

Usually, we would say lame. We would say that he is lame. It is an adjective. Lame is not always considered to be polite, so it may be better to say he walks with a limp or he limps.