When would you use penge and when pengene? Is pengene when you're talking about a specific amount of money and penge for money in general?
penge = money
pengene = the money
They are both plural in Danish.
No, "a coin" is "en mønt" and "the coin" is "mønten".
Both bills and coins, yes. Both is money (hope "is" is the right verb here)
I thought money was plural in both AmE and BrE. Are money always considered one 'lump', and not several units?
Money is definitely singular in BrE and as far as I know it is in AmE too. I guess you could say it's considered as a lump. The units would be pounds/dollars/whatever currency you use. There is a plural form 'monies' but it is only used in financial contexts, where it is used to emphasise that money comes from a number of different sources.
Money is uncontable in English, but countable in Danish. That's why they say mange penge, instead of meget penge! and many danes say "many money" when they speak English, it's a very common mistake