Depending on what other underlying condiitons are present, lawn mosses can survive quite well in full sun. Ask me how I know - lol !!

You will need to remove before time to overseed. I never bother with a moss killer as the stuff rakes up easily without it so that is just an extra and unnecessary step and expense.

It does depend on the soil, moss, and whatnot. If the soil's rich and well-drained and grass wants to grow and spread, the moss is not going to be able to compete with it and will die in short order. In poor soils with bad drainage, the moss will never leave because it's perfectly happy there and the grass isn't.

Moss killer doesn't make any sense either way, really. If it's going to die, it will. If it isn't going to die, it'll just come back again. Although, for a cheap moss killer, just spray iron or ammonium sulfate at about 4% solution on either one, it'll knock out moss with no problems.

I use that more on my patio and oddball corners where I don't want moss hanging out than anything, though, and ignore any that shows up in the north-east bits of lawn. That stuff is just going to stay there no matter what I do anyway.

We see success spreading lime over the lawn

LIme has no impact on moss.

Moss usually grows in wet areas and/or in highly acidic soils. The lime helps to bring down the acidity thus correcting the moss. Best to test first.

Moss can grow in acidic soils.......it also grows perfectly well in alkaline soils. Local soils are only slghtly acidic - 6.3 to 6.5 - so well above the suggested threshold for liming. But moss does thrive in many lawns here, generally due to lots of shade cast by native trees and by our damp fall, winters and springs.

The reasons for moss development are typically as follows:

excessive shade

excessive soil moisture/poor drainage

low soil fertility

soil compaction

In short, all the reasons turf grasses do not thrive. Improve the conditions to encourage a healthy and dense lawn growth and moss becomes only an incidental issue.

I wish moss would take over my yard!

Leaving moss intact is a personal choice option :-) Not everyone wants a moss based lawn and in many HOA situations, it is highly discouraged or even prohibited, just like any other lawn 'weed'.