Cut down on = To reduce the frequency or amount of DOING something. Usually it is structured as: "Cut down on (verb)ing"
Cut down = To reduce the amount of something, usually a noun (object). Usually it is structured as "Cut down the (noun)"
The council will try to cut down the annual budget.
Our family needs to cut down our intake of red meat.
My dad is trying to cut down on smoking.
Our boss is cutting down on taking breaks.
If you have more specific examples, I could address them. I wouldn't worry though, the difference is very small.
MATTER VS. COUNT
"Matter" means to be of importance in a general sense.
"Count" means to contribute or add towards a larger goal or total.
Taking a sick day does not count against my vacation time.
Taking a sick day doesn't matter if I get fired next week.
Kobe's shot was too late - it didn't count.
Kobe made the shot, but it didn't matter. The Lakers lost by 1 point anyway.
Sometimes, you can use either one.
Every minute matters in face of an emergency.
Every minute counts in face of an emergency.
Either of these is fine, but I would choose "counts". It emphasizes that total amount of time that one has in an emergency. But it is not strange to say "matters".
Student or Learner