Learn Swedish: Arbetsgruppen tror han är med och leder ekobyn. - The working group believes he is involved in leading the eco-village.

This is an example of participative pseudo-coordination. The phrase vara med och (to be involved in / to participate in) is followed by a verb that matches the tense of vara. Since är is in the present tense, the second verb must also be in the present tense: leder.

Would it be equally natural to translate "involved in leading" literally as involverad i att leda?

While involverad i att leda is grammatically correct, using the structure är med och + [finite verb] is significantly more idiomatic and common in everyday Swedish for expressing that someone is participating in an action.

Is a word missing between tror and han?

You might expect the subjunction att (that) here, creating the clause Arbetsgruppen tror att han.... However, just like in English, att is very frequently dropped in everyday spoken and written Swedish after common verbs of thinking or saying.

Why is tror used here instead of tycker for "believes"?

The verb tror is used when you express a belief, assumption, or guess about a fact (whether he leads the village or not is a matter of fact, even if they aren't 100% sure). Tycker is strictly used for expressing personal opinions or subjective evaluations.