Translation:She is buying a lot of different things.
I wouldn't say that, there's not much of a difference. Here, 'med' might direct the attention even more towards what is being bought, instead of the one purchasing.
'Med' is used in other sentences as well: -"Jeg er med dem" (I'm with them) -"Hun snakker med ham" (She talks with him) -"Hvordan går det med deg?" (How are you doing?)
Some more examples of 'med' (in Norwegian): http://www.nob-ordbok.uio.no/perl/ordbok.cgi?OPP=med&ant_bokmaal=5&ant_nynorsk=5&bokmaal=+&ordbok=begge
As far as I understand both translations, you can use both 'mengde' and 'masse' to express something like 'a lot of'. But 'masse' also translates to 'mass' (e.g. 1kg) and 'menge' refers to 'amount' (e.g. 1 mole, 10 pieces). Therefore it is not always interchangeble, especially in a scientific context!
Antall means specifically 'a number of' while en mengde means 'a lot of', or perhaps colloquially 'a whole bunch of'. So while they both have essentially the same meaning, this sentence isn't asking for the translation for 'a number of', so it's better to stick to en mengde here.