Learn French: Le jardinier vient de tondre la pelouse.

You would use the causative construction faire + infinitive. The sentence becomes Je viens de faire tondre la pelouse au jardinier. Notice that the gardener is now introduced by au (à + le). In causative sentences, if the action verb (tondre) already has a direct object (la pelouse), the person performing the action becomes an indirect object introduced by à.

Why does this sentence use vient de tondre instead of the passé composé (a tondu)?

The structure venir de followed by an infinitive is the recent past (passé récent). It specifically means that an action just happened. While a tondu simply means 'mowed', vient de tondre translates to 'just mowed'.

Can the verb tondre be used for cutting other things, like hair?

Tondre is used for mowing grass, shearing sheep, or using clippers to buzz off hair completely. If you are talking about getting a standard haircut, you should use the verb couper (to cut), often in a causative structure like se faire couper les cheveux (to get one's hair cut).

If I want to replace la pelouse with the pronoun la (it), where does it go in this sentence?