Learn Swedish: Misstanken är att någon måste ha lagt giftet i vinet. - The suspicion is that someone must have put the poison in the wine.

Yes! In Swedish, it is very common to drop the auxiliary verb ha in subordinate clauses (clauses starting with words like att). Saying att någon måste lagt giftet is perfectly natural and sounds very fluent, especially in spoken Swedish. The phrase måste ha lagt is an epistemic deduction, meaning you are making a strong logical conclusion about something that already happened.

Why is the supine verb lagt here instead of legat?

Lagt is the supine form of lägga (to put/lay something), which is a transitive verb, meaning it requires an object (here, giftet). If you used legat, which is the supine of ligga (to lie/be situated), you would be talking about the state of something resting, rather than the action of someone actively placing it.

I recognize gift as the word for "married". How does it mean "poison" here?

It is the exact same spelling, but functioning as a noun instead of an adjective! As an adjective, gift means "married". As an ett-noun (ett gift, definite giftet), it means "a poison" or "a toxin". The context makes it obvious here, as you cannot put "the married" into a glass of wine.

What is the root of the word misstanken?