(second try, the first one I accidentally deleted)
||to catch / capture
The stealing man runs
This is a general statement about stealing men: they run.
This is a statement about a particular stealing man who is running.
盗んだ is the past tense of the verb 盗む (to steal). Japanese does not have a participle, but we can use verbs, or indeed entire sentences, as adjectives by putting them in front of a noun.
The man buys a stolen vase
Here the vase is the thing that is stolen, so we use the passive form 盗まれる of the verb 盗む, once again in past tense to get 盗まれた.
The man bought a stolen vase
Same thing as above, but with the main verb 買う in past tense.
I catch a man stealing
This sentence is apparently unnatural, but we can’t quite figure out why…
盗んでいる is the continuing form of 盗む, thus 盗んでいる男 is the man that is currently stealing. I catch him (捕まえる).
I stopped the man who had stolen a vase
花瓶を盗んだ is a sentence meaning “to have stolen a vase”, which modifies the noun 男 to get “the man who had stolen a vase”.
The merchant will not buy the stolen vase
Japanese has no future tense, so this could also mean “The merchant does not buy the stolen vase”. It does not capture the meaning of “The merchant does not want to buy the stolen vase”, which would be:
I protect the vase that will be stolen
Since the vase is not stolen yet, it is uncertain. This is why であろう is added: 盗まれるであろう means “it will probably be stolen”, “it may be stolen” or “it will surely be stolen”.
I hit the man with a stolen vase
This depends on who stole the vase. If it’s a vase stolen by somebody, then 盗まれた花瓶 is correct. However, if I myself stole the vase, it should be 盗んだ花瓶.
Also, English is ambiguous here, so I put the main verb 打つ in present tense.
The man hid jewels inside the stolen vase
Once again the difference is between whether I stole the vase or somebody else did.