Do copula linking verbs show tense?

Yes, linking verbs can provide the context for tenses. The phrase "The eggs smell rotten..." would be present tense. If you changed the wording to "The eggs smelled rotten..." it would then be past tense. 

Of course. To be a sentence, a group of words MUST include a verb that carries tense. It isn't always obvious in English, but it's there. You can find the tense marker by changing the time reference, as Friendly Stranger showed you with smell and smelled.

Consider this example:

I can smell the stew cooking in the kitchen.

I could smell the stew when I walked in the door.

"can" changed to "could" which shows you that the modal auxiliary is carrying the tense in that sentence. When there's an auxiliary, "smell" or any other verb no longer carries the tense.