Learn Icelandic: Hvaða leið tökum við?

No, and that is great news for learners! Hvaða (which/what) is a special word in Icelandic because it never changes its ending. It stays exactly the same whether the noun after it is masculine, feminine, neuter, singular, or plural.

Why is there no word for 'do' in this sentence?

Icelandic does not use 'do' to make questions. Instead, you just swap the order and put the verb right after the question phrase. The structure here is literally 'Which route take we?'

Since we are taking the route, shouldn't leið be in the object form (accusative)?

It actually is! Because the route is the thing being taken, it is in the accusative case. However, for the feminine noun leið, the accusative form happens to be exactly the same as the basic dictionary form, so it looks unchanged.

Does tökum mean 'take' or 'are taking'?

It means both! Icelandic does not have a separate continuous tense like English does. The single verb form tökum covers 'take', 'are taking', and sometimes even 'will take' depending on the situation.