"Is there a lot of luggage?"
Weird question, but is there a reason for the use of は rather than が? My inclination is to translate this as 「たくさんにもつがありますか？」(also ommitting the の on account of it not bwing possessive?) I've always struggled with proper particle use-- but I've noticed Duolingo tends towards using は where I want to put が...
Wrong. たくさん can function both as an adverb and as an adjective. When used as an adjective, it's normally used with の. "たくさんの" is very commonly used in Japanese. You can say, for example:
~ Adjective function ~ たくさんの本を読みます。 "I read many books".
Of course you can also say: ~ Adverb function ~ 本をたくさん読みます。 or たくさん本を読みます。 "I read books a lot".
So yes, たくさん can be used with の when it functions as an adjective.
The form of this particular sentence, unlike with most other examples here, seems to be different in Japanese than it is in English. The literal translation to this one seems to be along the lines of "Is the luggage plentiful?". Meaning the sentence focuses less on whether a "plentiful luggage" exists there (あります), and moreso on whether the luggage is (です) "plentiful". It just sounds weird to phrase it that way in English, so instead it's phrased similarly to the other sentences.
It's along the lines of 花は白いです vs 白い花があります. It's like the case with です describes an object, while あります points out that an already described object exists (probs not a universal rule). Take this with a grain of salt, since I have no strong knowledge of these rules to refer to, more of a vague feel of the sentences.
it's like the difference between "a lot" and "many".
多い also has an implied "more than before" in it, and たくさん is more like "there is enough already", doesn't have to be "many", it just means that is more than usual or expected.
These two are very similar, the usage depends more on what you are saying, so it is not something you can understand without using the language.
If you want to know a bit more though, here is a thread where a native gives more examples of how they differ:
荷物が多いありますか and 荷物が多いですか are identical in meaning; in fact they are the same sentence with different omissions. です does not have an inherent meaning, it is an auxiliary verb that stands in when the verb can be inferred but you do not want to be so casual as to skip the verb, or when more conjugation is needed but the main verb is already conjugated (ありませんでした)
多い does not require あります, but uses です instead because it is an adjective. Also, "is the luggage a lot" wouldn't always mean the same thing, but it would actually depend a lot on context, like maybe a lot in price, in weight, possibly quantity, and more. If it was quantity, it certainly wouldn't sound as natural though.