Translation:Thank you very much!
I went on the Japanese Google and searched どもありがと, mainly references to the song and English translations. Another time I read online that only "old people" use it, maybe they meant people who were alive even the song came out. I did recently go to Japan and our tour guide told us to use どもありがと, but that might have just been to entertain the predominantly older crowd.
Putting どうも (doumo) in front of ありがとう (arigatou) gives you the meaning "very much", whereas adding ございます (gozaimasu) to the end makes it polite. Neither どうも nor ございます are used predominantly by old people. It's just a matter of how thankful you are and the level of politeness that you want to use. If you're speaking to someone who you don't know well or who is older than you, it's best to include ございます, which is for when something is happening right at that moment, like someone giving you a compliment. If something has already happened, like if someone took you out for dinner, you would use ございました (gozaimashita). Also, どうも can be used on its own (the word ありがとうis implied), but it's considered much more informal.
I may be wrong, but it seems that "domo" is just a polite word. I've seen it used in contexts other than just saying "thanks", however it still can be described as the word for "thanks" since it's the shortened version of "domo arigatou". You can also simply say "arigatou" but I think saying "domo" is a bit more natural, especially in a casual setting.
どうも→ very much ありがとう→ thank you (by itself, this is somewhere between casual and formal) ございます→ an honorific that makes the sentence more polite. The word ございます comes from ござる which means "to [honorifically] be/exist"
This sentence, if broken apart, would mean "thank you very much [honorifically exists]." It's used all the time here by people trying to show a lot of respect.
No, she's just speaking fast. (well, to new learners anyway)
She says the "mo" but because Japanese time their syllables evenly the following "a" comes faster than you might expect. I checked the audio and confirmed, but you can too.
Also it's been 6 months since your post so I suppose it could have been wrong back in beta but fixed now. I still think it's just the tempo of the words not having that usual space between them like in English that threw you (and others) off but maybe you were right.
You could say, "Thank you so much!" instead of "thank you a lot". You could also say, "Thank you very much!" which is the most used.
"Thank you a lot" is not good English.
EDIT: The downvote is not appreciated. :/ I usually don't care but thought I'd just point out that whether or not you like my opinion it doesn't change the fact that "thank you a lot" is incorrect English. The whole point of these forums is to learn, and I'm simply trying to help. I'm a native English speaker. It sounds weird to say "thank you a lot". You should explain why you are correct, that way I can learn. If you are wrong, you are fine. Now you know better.
NOTE: I see A-Big-Idiot (that is your profile name, not me calling names) is trying to learn Japanese through English. That is very difficult. I used short sentences to be understood easier. I am perfectly capable of using more advanced English sentence structures, but in good faith did what I could to assist in the comprehension of my words. If you'd like to discuss anything, feel free to reply below. That would be better than down-voting because you don't like my help. If you don't want to discuss then... Why are you here again?