It has to do with selections/masks really. By default on a normal layer/canvas etc you are on Draw Anywhere mode. When you create a Selection it switches to Draw Inside. You can then change this to Draw Outside, or Draw Anywhere (which effectively ignores the selection). You can see what setting you are on by looking at the Navigator Panel. The first icon underneath the preview image (looks like a cog wheel - is Open Navigator Settings), displays a drop down list. The last item there is Drawing Mode, which will show you what setting you are currently on by a tick against it.
Without a selection you wont notice anything happening.
If you want to only paint on paint on visible pixels that would be the Preserve Transparency button in the Layers Panel (first icon at the top-ish of the Panel - looks like a square with a lock). Unlike PS this locks all the layers, not just the exact ones you may want.... so you have to turn it on and off when needed etc - I don't think there is any keyboard shortcut for that.
Yes, a bit of a shame. Photoshop doesn't seem to lag when you make a selection and paint within the selection but Painter seems to lag when you paint with any soft brush during a selection.
I like to work on separate layers like 1 base colour layer, 1 shadow layer and 1 highlight layer (3 layers in total) and hitting the "preserve transparency" button doesn't exactly do what I want. That is I want to paint the shadows and highlights but within the boundary of my base colour layer however Preserve Transparency active on empty layers prevents me from painting anything other than the base colour layer.
I've also tried wacky things like going Layer > Create Layer Mask From Transparency on the base colour layer, then duplicating the layer, deleting the content but leaving the mask, this somewhat decent as I don't have the lag but the duplicate layer seems to have 1 pixel ring around the base colour shape.
I really enjoy the feel of painting using Painter but just a number of quirks getting in the way that I wish was gone. It seems Painter was intended to simulate the workflow of traditional painting Ala Prima style in addition to the natural brush effects. It doesn't seem to work in the way of layers and selection marquees, every brush stroke you paint is expected to be painted on top of what was there on the same layer.