I would use the perfect. I HAVE started...

Both are commonly used when speaking, however in writing it is more grammatically correct to write, "I've started to cut down recently." The reason being is because "started" and "recently" is past tense, hence "I've" which is a conjunction for "I have" is also past tense, as in you have already begun to cut down.

Yes both are commonly used "I have started to cut down recently" is the correct way.

I would probably use "cut back" rather than "cut down" But I agree with everyone else as to the tense.

yes.i agree to yours