Stone compaction on placement

You'll need to remove the old mortar - you can't just slap on some new mortar and expect it to hold if the old stuff hasn't been removed. Chisel and brush with a stiff wire brush.

No form should be needed - if you appear to need a form, you need to add more dry mortar mix to your bucket - preferably, don't add too much water in the first place. Consider a brick wall - there's no form on that. The stuff should look a bit too dry when you mix it, but wet up when smacked around with a trowel. If it looks wet when mixed, it's usually too wet.

If you have actual mortar rather than "sand mix" there's a degree of self-levelling from the small stones in the mix - they are trapped between the base and the step, and set a minimum thickness for the joint. Normally you want exterior steps just a hair off level so water drains off them, but not so much as to make them hazardous to step on. A level and a mallet (or a hammer on a block of wood) will get you there. If you run out of adjustment, pull the step up and add more mortar where needed, then smack it back down.

Weather matters - if the mortar freezes before it cures, it will fail. A quick-fix now may need to be redone next spring in order to hold for the long-term - or, you buy a bale of straw and bury the step before the cold front comes through - or make a plastic bubble and run a heater in it.

Modifying a dry stone wall is a tricky operation. Normally you need two kinds of chisels, a pointing tool and a pitching tool. You can buy them from a masonry supplier like Micon. I would get PKM-19 and SM40-R. Get a mallet too (1031-A).

Grave a deep furrow where you want to break off the stone, set the pitching tool in the furrow, then give it whack. If everything goes well, the stone will split clean off. There is a certain amount of feel that is required. Making sure the furrow is nice and deep will help ensure success.

Why you don't want to use an angle grinder. (In response to comment) The immediate problem with a grinder is that it will make a very unnaturally smooth face that will look nothing like the faces of the other stones, so the ones you cut off will stick out like a sore thumb and look even worse than they do now. The other reason is that cutting rock with an angle grinder, especially indoors, is a bad idea at multiple levels. First of all, if the rock is somewhat hard, like slate, it will take forever. You could have a diamond blade and be standing there cutting for 15 minutes and make it through 1" of stone maybe. Meanwhile you will be surrounded by plumes of dust and toxic fumes as shards of slate go flying in all directions at literally bullet-like speeds.