New lawn in need of rescue

I didn't realise you'd done any mowing because it looked so dead!
If you're just trimming lightly, that's ideal - little and often is always the best method as it encourages the grass to spread, and will help to fill spaces over time.
Normally, you wouldn't cut too soon after getting a new lawn laid, or sown, but if it's managing, I can't see it being a problem. It's also to prevent the ground being compacted too much, so it's always better to avoid being on it for a good length of time after being laid/sown. In wetter, softer ground, you can get more dips and hollows too, but I'm guessing that's not the case with yours.
Ours doesn't grow anything like as quickly as that at this time of year, despite the rain, because it isn't as hot here. It's always about working with your conditions though, regardless of the type of plant - and that includes grass.