Is it considered rude language to omit interrogatives in speech?

Solution 1:

What you are talking about is totally sound in terms of grammar; it's just informal. It's not generally considered rude at all, except on occasions using informal language in general can come off as rude. (Of course, your last example with "what the _" has the potential to be quite offensive, but that would be determined by the profanity you decided to insert into the sentence and not whether you dropped a verb.)

This process is systematic. You are omitting either the copula or the auxiliary verb in an interrogative sentence. This means the word that is dropped is either be, have, or do.

• (Are) you tired? (copula)
• (Is) he coming with us? (auxiliary)
• (Have) you seen anything like this before?
• (Does) anyone want tea?

People do this all the time; probably more than they realize.

As for "(what) the hell" and such constructions, the dropping of what and the syntax behind it is a bit of a different animal. Profanity often has extra-linguistic properties; even the "what the _" construction (forgetting about omitting what) itself does not follow any kind of normal syntactic structure at all.

(For another example, check out this extremely interesting linguistics paper: English Sentences without Overt Grammatical Subjects. Warning: this paper examines profanity so... don't be surprised at what you read.)

Solution 2:

It is not rude by itself, and you can certainly find it in regular usage, esp. in spoken language e.g.

Examples (even shorter / OP's / full form, formal):

Q: Still there? / You still there? / Are you still there?
A: Yeah. / Yeah. / Yes, I am here.

which is perfectly understandable and not rude at all.

It is informal, (and I would say on the same level as the answer 'Yeah' is) so in situations where being informal is not appropriate it would be inappropriate.

With your other example

There a doctor in the room?

the shortening achieved by omitting the "Is" is so small that, in my opinion, you will not find this example nowhere nearly as often (and probably only as a slang that emphasizes such constructs).