Potentially. Their larvae are Leatherjackets and they do eat grass roots. Grass literally dies off in patches. You can treat the lawn to kill them. I suppose mowing the adults will reduce the population.

Over the past 2-3 days I have seen quite a few lurking around. I think it must be the season for the adults to come out and mate.

They won't harm your trees, they're primarily a turf pest, and can damage lawns when they're larvae (leatherjackets), but most of the damage, if any, is birds pecking at the turf to get the larvae out. If you saw hundreds of flying ones, the larvae have hatched out and gone. You may want to consider a treatment for the lawn in the form of nematodes, which are watered on at the right time of year. Most are temperature dependent - these are available from Nemesys online, but make sure you order the one for leatherjackets. It may be too late to use it this year, not sure.

I'd wait until next year now - no use trying to kill them when they've left.

They've left, yes - but not before laying more eggs, Steragram. They usually mate before their first flight... and laying eggs might be what they were doing when disturbed by the mower...

Would the nematodes eat the eggs?

Not sure - just checked and the information says to use the treatment two weeks after seeing adults near the lawn, but only if the temperature is above 12 Deg C, and the ground is moist - if it isn't moist, it needs to be kept so for a week or two. Eggs hatch after two weeks, so the nematodes must target the larvae rather than the eggs.

Well, since you've mowed your lawn they will be daddy short legs now. LOL.