Learn Icelandic: Við höldum það ekki.

In Icelandic, halda is commonly used to mean "to think" in the sense of having an opinion or believing something, much like saying "I hold that opinion" in English. When you want to say "I think so" or "I don't think so", you use halda.

Since halda is a strong verb, shouldn't the vowel change to "e" like in ég held (I think)?

The stem vowel change to "e" (like in ég held or hann heldur) only happens in the singular forms. In the plural, strong verbs usually keep their original stem vowel. However, because of the -um ending in við höldum, the "a" changes to "ö". This is a regular plural change, not the special strong verb singular change you are currently learning.

The English translation says "We do not think so", but isn't það the word for "it" or "that"?

Exactly! A word-for-word translation would be "We think it not." In Icelandic, you don't say "think so"; instead, you refer back to the idea with the pronoun það (it or that).

Why does ekki come at the very end of the sentence?

In a simple sentence, ekki usually comes right after the verb. However, when the object is a short pronoun like það, that pronoun steps in front of ekki. So instead of við höldum ekki það, it correctly becomes við höldum það ekki.