Learn Hardcore Dutch: Met die museumkaart zouden we alle tentoonstellingen kunnen zien. - With that museum card we could see all the exhibitions.

Because the sentence begins with the fronted adverbial met die museumkaart, Dutch word order in main clauses places the finite verb in second position. As soon as something other than the subject is first, the verb comes next and the subject follows.
Example without fronting: We zouden met die museumkaart alle tentoonstellingen kunnen zien.

What happens to the word order if I don’t start with met die museumkaart?

If you put the subject first, you follow standard S-V-O-… structure. For example:
• We (subject)
• zouden (finite verb)
• met die museumkaart (adverbial)
• alle tentoonstellingen (object)
• kunnen zien (infinitives at the end)
→ We zouden met die museumkaart alle tentoonstellingen kunnen zien.

Why is it die museumkaart and not just de museumkaart or deze museumkaart?

• de museumkaart = “the museum card” (neutral, definite)
• deze museumkaart = “this museum card” (something close or just introduced)
• die museumkaart = “that museum card” (referring back to a card already mentioned or known, or something more distant)
Here die is a demonstrative adjective pointing to a specific, previously mentioned card.

Why is it alle tentoonstellingen rather than elke tentoonstelling?

• alle + plural noun expresses the entire set collectively (“all the exhibitions”).
• elke + singular noun focuses on each individual item (“each exhibition”).
Because the idea is to see every exhibition as a whole, Dutch uses alle tentoonstellingen.

How do the verbs zouden, kunnen, and zien work together, and why do the infinitives end up at the end?

Dutch allows “verb clusters.” Here:

zouden = finite auxiliary, conditional of zullen (“would”)
kunnen = non-finite modal (“be able to”)
zien = non-finite main verb (“see”)
Rule: in main clauses the finite verb sits in V2, while all non-finite verbs (infinitives/participles) go to the sentence’s end in the order they appear in the bracketed underlying structure: [zouden [we [kunnen [zien]]]].

Could I use mogen or mochten instead of kunnen here?

No—kunnen expresses ability or the possibility to do something. mogen (present) and mochten (past/conditional) express permission granted by someone or something. A museum card gives you the ability to enter and see exhibitions, not special permission from a person, so kunnen is the correct choice.

How would I turn this statement into a question?

Invert the finite verb and subject for a yes-no question:
Zouden we met die museumkaart alle tentoonstellingen kunnen zien?
You can also keep the adverbial first and use rising intonation in speech:
Met die museumkaart zouden we alle tentoonstellingen kunnen zien?

Can I drop the article and say “Met museumkaart zouden we alle tentoonstellingen kunnen zien”?

No. In Dutch, singular common nouns normally require a determiner (definite or indefinite). You cannot omit it: you need de, die, een, etc. So you must say Met de museumkaart or Met die museumkaart, never Met museumkaart.