Cutting height and re-seeding for a particular mix

That height should be fine. When in doubt, it's always better to err on the side of higher. If you're just going to reseed some spots that didn't take, you should use the same or similar mix. If you use something different, then you can wind up with obvious differences, visible, where one area has a different color, texture, grows faster, etc.

It seems to me that you have chosen a good general purpose seed mixture which should perform well and create a good quality lawn.

I would say you are mowing a bit too high and would recommend a height of 3-4 cm once established. The key to a healthy lawn is regular mowing, at least once or twice per week and remember to make sure your blade is sharp!

In cool, damp climates, close mowing works just fine. It provides a more manicured/finished look. It keeps the turf drier, which reduces the risk of fungal diseases, but it does tend to encourage weed seeds to sprout and grow.

Here in the midwestern US, we generally mow short in cool damp weather and tall in hot or dry weather.