Yes, it is definitely an understatement: it means “I would REALLY like to do that, but I won’t/I can’t/I have a good reason not to”.

Some examples I can think of:
“Peux-tu venir chez moi demain soir ?
-désolé, ce n’est pas l’envie qui m’en manque, mais je dois travailler”

Or, imagine a parent talking to a child who has done something wrong:
“je ne vais pas te punir, mais ce n’est pas l’envie qui m’en manque, tu le mériterais”.

For the second question: “ce n’est pas l’envie qui manque” can still be understood, but I personally think it is much less common (and maybe a bit more formal ?).

The “en” refers indeed to whatever the “envie” is for. In the 1st example above, “en” would be for “going to your place” (ie, “ce n’est pas l’envie de venir chez toi qui me manque”).

For what it’s worth, I’ll only add that the more technical term is litote, that is, expressing something by negating the opposite, and litotes are generally considered to be understatements by definition.

While this may look like an understatement (as per Circeus answer’s, it’s an understatement by definition), it feels more like a lukewarm willingness to do the thing. To me it’d say "I’d be down for it if there wasn’t this minor thing that I’m unwilling to report/get around".

e.g.

Ça n’est pas l’envie qui me manque, mais j’ai piscine ce jour-là.

C’est pas l’envie qui me manque, mais je dois me coucher tôt.

As for the second question, "m’en" sounds bloated to my ear, whereas "me" sounds much more natural, and is still grammatically correct. You can therefore drop just the "en" or even "me" as Greg said.