Kevin told you a very convenient alternative way to order something:
[Thing to order][quantity + TSU or counter suffix]、onegai shimasu.
Then I’d like to answer your question one by one:
■would it sound rude if you said "watashi wa udon" to a waiter?
If you are **not the first person** to order at your table, I think it’s not rude at all, natural and casual.
E.g. Takeshi: (to a waiter) Oyakodon o kudasai.
You: Watashi wa udon.
If you are the first person, it sounds a bit curt or curious, because it could sounds like “I am udon,” rather than “I’d have udon,” at this timing.
■If so, should you use "udon o kudasai" or "udon kudasai" ?
These phrases are very common. "Udon o kudasai" is a bit politer.<(very) formal> Udon wo itadake masuka.
I think you don’t have to use this at a Udon noodle restaurant.
■Could you just point at the menu and say "kore kudasai/ kore o kudasai"?
Yes, you can. I think it’s a good way.
■And if I put a "desu" at the ending of Mary`s sentece, would it mean "I am udon"?- just curious ^^
You’re right. A bit curious ^o^ If you’d like add something, “Watashi ha udon **o**” , just cut “kudasai”, would be fine.
Hope this would be helpful.