Yes, you're right about 我的名字叫 being more formal/polite, because it's more complete. It's actually equally good to introduce yourself saying 我叫 Meir or 我的名字叫 Meir. I'd say they're interchangeable.
You can ask what the other person's name is by saying
你 叫 什麼名字?or 你 叫 什麼?
However, the latter will sound rude if it's not said in a rather upbeat tone and is almost always said after introducing yourself first.
You can make it more polite by adding a 呢 (ne) to the end and it becomes 你 叫 什麼名字 呢?or 你 叫 什麼 呢? (Please use 吗 for a Y/N question and 呢 for a Wh- question.)
But the latter will still sound a bit rude.
As for the "and" between two clauses, please don't translate it. The English word "and" doesn't work the same in Mandarin. The word "and" is translated into 和 almost only when it's a juxtaposition of two nouns.
For example,
我(wo3)喜欢(xi3 huan1)电影(dian4 ying3) 和(he2) 运动(yun4 don4)
= I like movies and sports.
(With a juxtaposition of two adjectives/adverbs, it's another story. We use 又(yo4) in the "又 adjective/adverb 又 adjective/adverb" structure. Therefore, "I'm tall and handsome" will be "我 又高(gao1; tall) 又帥(shuai4; handsome)".)
As Cherry said, 我是美国人的 is the literal translation of "I'm American". Much as it's correct, it's not very natural. When talking about nationalities, English speakers tend to use an adjective (I'm American/Canadian/Indian/Australian...), but Mandarin speakers will almost always use a noun, in the pattern of "country + person/people (人; ren2)".
We say 我是美国人/加拿大人/印度人/澳洲人; literally, I'm an American/Canadian/Indian/Australian person.
I think it's okay to translate "I'm from the United States" into 我是美国人, despite the fact that it's not the literal meaning, which is "我是美国來的", which in Mandarin is very colloquial.
In my opinion, to introduce your name and nationality, the formal/polite version is "我的名字叫 Meir。我是美国人。" To go colloquial, say "我叫 Meir。我是美国來的。"