Yes, that can be another correct translation when vouloir is used as a semi-auxiliary verb (as it is above). However, note that "would like to" is an idiomatic semi-auxiliary in English, so if you're translating from English to French, you should translate the entire thing as vouloir instead of translating each word separately.
Snofferol, "manger" can be translated to "eat" or "to eat" whereas "mangeons" can be translated to either "eat" or "are eating".
As it is not grammatically correct to write "Nous voulons mangeons le poulet." in French, and the literal translations of this incorrect French sentence ("We want eat the chicken", "We want are eating the chicken.") to English are also grammatically unsound, we therefore have to use "manger" instead of "mangeons" in "Nous voulons manger le poulet.".
Another easy way to remember is whenever you see a modal verb ("voulons", "pouvons", "doivons"), this modal verb is almost always followed by a verb in its infinitive form (e.g. manger, boire).
Yes, it is the politer way much like English.
- Je voudrais une bière
Is politely asking for a beer. Note voudrai is actually "I will want."
From what I've seen je voudrais is also interchangeable with j'amerais - which more literally means "I would like" rather than "I would want." but je voudrais seems more popular.