Is it right to say "a hurt leg"?

It is not totally perfectly correct. In the dictionary, 'hurt' has three meanings, all as an active verb (not a passive verb). You can hurt someone else or yourself (active verb) but you can't be hurt (passive). So you can say your leg hurts, but you can't say you have hurt leg.

But the way we speak, the vernacular, it's perfectly acceptable. If you said you had a hurt leg, everyone knows what you mean.

Generally, in Britain at least, you would refer to an "injured leg" or an "injured arm" or other part of the body.

A person can have "hurt feelings" (emotional stress).

You could, but it would be better to say either "the leg hurts" or "there is an injured leg."

As far as I know, "hurt" is an adjective that describes one's feeling, so I want to know if it's able to use "hurt" like that.

It depends on the leg part. For instance you could say "a busted knee", "trick knee", "funny ankle"...

yes, but it's more natural to say i;ve hurt my leg.

depends on how you use it.