If it's a typical cool season grass mix that we're talking about, I'd
cut it about 3", which should leave it looking very nice and help keep
out weeds. The key word here is "help." You're still going to have
some. Since you currently have a lot, I'd use a herbicide to get rid
of them, which is very easy with dandelions. If you can spot treat
them, which is most effective and uses the least herbicide, get a back
pack sprayer. If there are too many to do that, then I'd use a weed n
feed product. Once you get rid of most of them, if you properly care
for the lawn, you can spot treat them from then on.
As for the neighbors lawn, you may not see the dandelion flowers
because they are mowing it frequently and short. If that's the case,
they can still be there along with other weeds, crowding out the grass
and making it look like hell.