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.

The object o kudasai form is also a correct way of ordering food. Another way of ordering is to say biiruu ippon onegaishimasu (One beer, please) So I guess that's another way of replacing the word kudasai.