Learn German: Er scheint sich extrem verdächtig zu verhalten. - He seems to behave extremely suspiciously.

The conjugated verb scheint takes the normal second position in the sentence. The action that he seems to be doing goes to the very end as an infinitive (verhalten), with the word zu placed immediately before it. This perfectly mirrors the English 'seems to behave'.

Why does the reflexive pronoun 'sich' come right after 'scheint' instead of later with the infinitive?

The infinitive verb here is the reflexive sich verhalten (to behave). Even though sich belongs to the infinitive action, standard German word order places pronouns as early as possible. Therefore, sich jumps up to the main clause right after the conjugated verb scheint.

Does 'verdächtig' mean he is feeling suspicious, or that his actions are causing suspicion?

It means his actions are causing suspicion. If you wanted to say that he is feeling suspicious or distrustful of someone else, you would use a different word entirely, like misstrauisch.

Why is there no special adverb ending on 'verdächtig' to make it mean 'suspiciously'?

Unlike English, which usually adds '-ly' to turn an adjective into an adverb, German almost always uses the exact same base form for both. Here, verdächtig acts as an adverb simply because of its position modifying the verb verhalten.