Much and Many are used in negative sentences and with questions. A lot and a lot of are used in affirmative (positive) sentences.
You dance a lot with... You don't dance much with... Do you dance much...?
I have a lot of french fries. I don't have many french fries. How many french fries do you have?
Much is used with singular uncountable things. Many is used with plural uncountable things.
How much time do you have? I don't have much time. How many hours do you work? I haven't worked many hours.
A lot of is used for both singular and plural uncountable things.
How much time do you have? I have a lot of time. How many hours do you work? I work a lot of hours.
If you are native speaker or REALLY fluent, you can get away with, under limited circumstances, breaking these rules but do it wrong and you will just sound ignorant or foreign.
You don't use "much" unless you are talking about a negative. Like "you don't dance much with your girlfriend" "I don't dance much anymore" "Much of my dancing is more appropriate for a wedding than a ball". "A lot" is more widely correct, you can substitute it for any of the "much" in the previous sentences and more.
I think 'a menudo' would be better to use for 'often'. In this case we've simply used 'mucho' which generally means 'a lot' or 'much'. Sure one could argue that these are all synonyms of each other. But there are some contextual differences in usage. I think to reflect that it would be better to keep a distinction between them.
Person Note: I think if one does have a girlfriend one should dance with her often. Just saying.