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I thought that it was considered rude to say as if we know that someone wants
something. So what would be the proper way in Japanese to ask if they want us
to do something? Like "Would you like me to tell you my number?"

Below is what I have in mind, please correct me if I'm wrong so I can learn
from it...:

  1. > 先生、私の電話番号を教えたいですか

This is wrong since we cannot use たい for other people...

  2. > 先生、私の電話番号を教えてほしいですか

This is wrong since we cannot use 欲しい for other people too...

  3. > A. 先生、私の電話番号を教えられることがいいですか？  
>  B. 先生、私の電話番号を教えることがいいですか？

I'm not sure which one is correct or if both are wrong. But even if it is
correct, it would also mean somewhat like "Teacher, is it okay if I tell you
my number?" and a bit different to "Teacher, would you like me to tell you my
number?"...

  4. > 先生、私の電話番号を教えましょうか？

I'm not completely sure, but this should means "Teacher, should I tell you my
number?". Although it has a difference nuance from "Would you **want me** to
tell you something?" but it serves a similar purpose. Is this the correct one?

_Additional for information purpose_. Based on Oal's comment, it seems it's
safe to use たほうがいい. as in

  5. > 先生、私の電話番号を教えたほうがいいですか？
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grammar
How to ask someone if they would like us to do something?
The most appropriate way to phrase the question will depend greatly on the
situation. It will change with familiarity, social status, and the context of
the conversation (i.e. is it a business meeting or are you asking a friend if
they want you to give him a lift to a party).

In general, ～ましょうか／～ようか , your No. 4, can be used without causing offence or
being overly formal.

For example: お[撮]{と}りしましょうか？ ( _lit._ Shall I take (your photo)) would be a
perfectly acceptable way of asking a stranger if they would like you to take
their photo at a location where that would be an acceptable thing to do. Think
asking a couple if they'd like you to photograph both of them with their
camera at a sightseeing spot.

In the same way, [手伝]{てつだ}おうか？ would be a common response to a friend telling
you they were moving house on Sunday. Close to the English, "Need a hand?"

For your specific example, I would probably use No. 4 exactly as you have it,
or 先生、私の電話番号を教えた方がいいですか？ if I wanted to nuance it closer to "Would it be a
good idea if I gave you may number."

No. 1 is wrong, as it would be asking if your teacher wanted to tell you your
number. But note that (again depending on circumstances) ～たい can be used to
ask what someone would like to do themselves.

No. 3 would be highly unnatural, although 3A is not strictly 'wrong'.

As oals commented, No. 2 would be acceptable in informal, friendly
conversation, but shouldn't be used to a teacher.
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answer