Seeded Lawn.....filling in patches

We had our lawn dug up, removed the grass, raked levelled etc and reseeded by the local Bowls Greenkeeper.

When the seed went down it seemed fairly evenly spread but as expected not all of it germinated.

When most of the grass established I started putting more seed down on the odd bare patch and where it didnt seem to have taken at all on one edge. I was hoping that the new seed would establish before I would need to cut the original seed.

Howeber now the lawn looks overgrown and the bare patches are partly covered by the long floppy grass! I want to cut the lawn on the highest setting on my mower, but am worried that treading over the seed that hasnt germinated may damage it, or even whether when i run the mower over the grass, the grass collector may suck up some seed - although presumably it doesnt work like a hoover in that sense!!

os there any danger in leaving the established grass grow more....will it starve some of the grass of light etc? Also there are a few broadleaved weeds on the grass (not half as many as before)... Should i try and dig these out by the root and patch the lawn again with more seed or use a weedkiller that only targets broadleaved plants, although guess most dont kill down to the root?

Thanks for any advice.