Do you want your plants to do well or do you want a particular appearance.

I used fine hardwood from my compost facility. A shovel pitched or sort of broadcast will cover the surface and look new then again when the appearance offends your taste.

I always tried to use mulch in such a way that it is NOT the thing, unnoticed. I personally hate the bright red mulches which to me look like an old woman with bright red lipstick.

My taste is mixed hardwood because it will absorb sticks, leaves, etc without those things being noticed. I never pick up a leaf, small limb or branch that falls, hoping for a natural look. And, as it works out for the plants this variety of organic matter improves the soil better than a particular mulch such as cypress or pine mulch.

I repeat, MY TASTE!

Depends in part on what it is you are mulching. Vegetable gardens have different wants than ornamental shrubs, etc., etc. I tend to favor leaving the old mulch where it is but my experience, such as it is and aint, is more with poor-man's eco-restoration and not with food gardening. Others will know better than I. I like to use wood chips that have been left alone in a pile for a year or more as mulch for trees, shrubs, and some flower beds.

"Do I need to remove last year's mulch before putting down the new stuff?" No. There is no need to remove last years mulch before adding more. Most of the time, like now as I am mulching the beds, what I put down last year has been digested by the Soil Food Web, but I have never had any problem adding more mulch on top of what might still be there. Ma Nature does that all the time.
I cannot be sure, but this makes sense to me, removing the previous mulch might also remove some of the SFW that is in that mulch working away.

Because I use primarily shredded leaves for mulch, it decomposes fairly quickly, I rarely have any "old" mulch to remove.

But as repeated here, NO, its not necessary to remove last years mulch...

I have seen dried up, compacted, old mulch form an hard crust that could block water flow. I don't think it should be removed but it needs to be broken up. Then new mulch can be placed on top. I use shredded leaves so it doesn't happen to me. But, I notice it in my neighbor's yards especially when they pile it thick to make their tree volcanos. There are so many of those around here...sigh :(