Lawn repair by newbie

Hello everyone. Moved in to a newly build property in Hampshire just over 2 years ago. Never been happy with the state of the lawn as grass was the wrong type for us, fast growing type, and also lawn was a bit bumpy. As a total newbie to the world of gardening I thought it's best to get a proper gardener to sort it out last year but the result was not was better than what we had in the first place, I think he just used some top dressing and seeds.

So been educating myself on lawns and have decided to give it a makeover. My plan is to cut short, rake, aerate using hollowtine & rake it off, top dress, flatten using back of a heavy steel rake & brush in, seed & use a water roller to press seeds in and flatten, fertilize. We have a dense chalky soil in the area, the lawn seems to have a topsoil of perhaps 2 to 3 inch deep which hold moisture well but I fear it doesn't drain well and feels pretty compact, PH is about 7 ... could be farm manure.

Q1-Some suggest that top dressing should be done after seeding but as my patch is a bit bumpy I worry that some seeds will be burried under 1 or 2 cm of top dressing, what do you guys think?

Q2- I don't think I have much of an issue with miss and thatch and also it seems to be the wrong season to scarify so thought it's best to skip this process and reconsider at at of summer?

Q3- What would be the best top dressing mixture, Local supplier suggested 50/50 as he said they mainly use 70/30 for sport grounds but I think more sand might be better given the compact nature of current top soil. Any opinions?

Q4- I am leaning towards overseeding with just slow growing seed mixtures, I would appreciate recommendations on suppliers and type?

Will try to attach some pics

 

Being on chalk myself, I don't think you have too much of an issue with compaction or drainage, but the aeration will sort that out if you think there is.
A quick summary would be
Hollow tine
Overseed heavily. You're feeding the birds as well
A quick scarify
top dress 50/50. Too much sand on chalk is a nightmare to keep watered during dry periods
To get a lawn looking fantastic, it takes a lot of work so be prepared to do it again after summer and next spring.
Grass seed gets right into the holes left by hollow tines, so the idea of burying them too deep isn't an issue, even with a top dressing( you're filling in depressions, not covering the whole area).
Scarifying will distribute the seed better and remove or break up the cores.