"Have you ever seen that show?" is standard, while "did you ever see that show?" is a common variant in American English (the adverb "ever" makes the meaning clear). Note that "did you ever see that show?" may be correct, if you are thinking of a time-frame in the past that does not come up to the present. For example, "did you ever see that show before it closed?" Also note that Americans often say "I just (ate)" instead of the standard "I've just (eaten)". This is potentially ambiguous, as "just" may be an aspect marker, or it may just mean "only" / "merely".

Variant with 'have' (Present Perfect) is correct. However, I suppose the Americans may use the first, because Present Perfect is sometimes replaced with Past Simple in American English.

"Have you ever seen that show." Is correct. Have you is usually used in the second person. So is the case with did you. Have you is sometimes used independently as a single expression following the main sentence. Did you is used along with a verb only Did refers to something done.

In this context, the ever in American English suggests an indefinite time, including the past. Did you see the game last night? (game is over, recently watched) Did you ever see the show Dallas? (show is over, watched in the past) Did you ever see the game last night? doesn't make sense because the game has a defined, recent time Did you ever see this season of the Bachelor? doesn't make sense because this is ongoing and watching needs to be recent. This does make sense once this season is over. Did you ever see that game from last season? The game has a defined time, but you could have watched it at any point.

Did you ever means the show was in the past, in a time period that is closed. The ever is not grammatically neccesary and is an emphatic for at any point in your life, probably in the past and not something you would be watching currently. Did you ever see the first season of the Bachelor? Did you ever see the original Rent cast? Did you ever see the show Dallas? You can use either version for a show that is current, but it would not make sense to say did you for a show that is still ongoing. Have you ever seen the Bachelor? (old or current seasons)