I get the impression you could do this. I am landscape architect, not a mason, and have dabbled a bit in masonry for my own projects, not for clients. At the beginning I pondered the same questions that mystify you now. If you have the expectation of perfection, it will be hard to achieve that as a novice. But you can achieve pretty decent results. If you lower your standards a bit, it will probably work out OK. :-)
For this repair, you'll need to remove as much of the old mortar as possible ... not so it is absolutely clean, but so it is sound and so there is adequate cavity to receive the new mortar (no less than 1/4" thickness anywhere -- more is better. If there is not at least that much cavity, the stone will stick up too high. Let no mortar remain that is not sound. Remove it all if you can as it will make applying the new mortar easier. Don't worry about any residue, though, as long as it is well adhered.
You'll want to use plain, common mortar. Portland Cement is the active ingredient in mortar. You wouldn't want to buy cement, as in order to make mortar, you'd also have to buy mason's (fine) sand, and whatever else is in mortar, separately. And this is a small job, so just buy a bag of common mortar mix. You don't have to mix the whole bag. Just mix a little more than what you think you'll need to use. It will be useful to have some dry mortar on standby to adjust the mix if you get it too wet.
If you don't get the mix moist enough, it will be impossible to use. It needs to end up being like soft butter that you can push around easily with a tool, without it "cracking" but not like batter. Imagine that a blob of it should hold a shape like a hershey's kiss without turning into a pancake. But if it is too dry -- like oatmeal cookie dough -- it will not stick to the work or allow adjustment of the work while it is being placed.
Time of setting the work is a factor. When you apply the mortar to the substrate, it is buttery. But very quickly some water is absorbed from the mortar into the substrate and/or, the work. Within seconds, the mortar can dry out enough that you can no longer work the stone that you're trying to set, down into it. Usually, one wets the substrate and the stone a little first to minimize this. It could be with a light mist or brush the water on (sloppy ... like a mason. :-) If you're setting the stone and the mortar dries too much before you can get the stone correctly situated, scape all the mortar off, retemper it (with a little water) and try again. (You can retemper freshly mixed mortar that is just drying, but you can't retemper mortar that has begun to set up chemically. You usually have about a half hour of working time, but this varies on account of temp.)
I still find it hard to create nicely finished exposed joints while working on setting the stone. My solution is to clean out all the joints and do them separately, using a grout bag, after the work has set up enough. This requires a slightly runnier mix ... just runny enough to run through the bag. Unmixed mortar that has been setting around in bags for a few weeks sometimes doesn't want to flow in a grout bag, in which case you just have to buy fresh.
You'll need a finishing tool to make the concave joint. If you use a grout bag, there will be a surplus of mortar in the joint. Let it set up some so it is no longer runny or messy. It might be like an almost baked butter cookie. Strike off the excess flush with the stone surface, using a standard mason's trowel. Then come back and press down the joint with the finishing spoon. Sometimes I don't want to spend 15 or 20 dollars on a tool for a little job. I look in the kitchen and see what will work ... though it will probably be destroyed.
If you've ever laid tile and know how the notched trowels create lines of grout on which to set the tiles, you'll either need a large notch trowel like 3/4" x 3/4" (or similar), or you'll have to create that effect in a homespun way with a regular trowel. When you set the stone on the fresh mortar, if the mortar is placed like a solid "slab" of it, there is absolutely nowhere for the mortar to squeeze out if it is too high. Only a little can come out of the edges of the work. Consequently the stone would be set too high. If the mortar is applied with a notched trowel, while each line of it is too high before you place the stone, each line of it has a cavity running alongside it, into which it can squish, allowing you to tap down the stone to the correct height. Have a rubber mallet and 2x4 ready before you begin the work. Place the 2x4 on edge across the stone and tap on it so you can move the whole stone in plane and don't take a chance on cracking it. You can run the 2x4 on to the adjacent stones as an aid in matching its height.
Try to work clean so cleanup is not a nightmare. Sometimes, stray mortar is easier to clean after it has set a little and can be knocked off and then wire brushed. But don't let any go too long or it will require acid.
You might have to waste some mortar trying to get things right, but that's cheap learning.
Good luck!
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