Ok...where to start. We lay natural and quarry cut stone all the time and hand point the mortarted joints.
Firstly, with your "grout" problem are you talking about mortar??. Where I come from grout is the stuff you put between ceramic tiles in your bathroom and mortar is put between bricks and blocks.
If so; and you are serious about fixing it you need to grind and chip it out. I have several special mason hammers (Estwing) for chipping out old mortar. Sometimes I use a diamong wheel on a 5" grinder or a Hilti wall chaser set on the width of the perp.
2nd, you can re-mortar or use a commercial grout repair product normally sold as a mortar repair or crack repair product. These products are flexible and available in a range of colours......but it seems often not exactly the same colour

It is also harder to get the same textured finish as other existing mortar. We always use a mild bleach to remove any mould first, then pressure clean, grout repair, then pressure clean again.
When mortar joints are uneven, as with like 99% of natural stone the tools of choice are wire brushes and your pinky....with a glove of course. There are a huge number of ways you can point a wall. We apply excess mortar to the joint, wait for it to go firm and then start work. You can just flush the joint by cutting with a trowel and wire brushing the stone to remove the excess. You can flush the join and "line" with a shaped piece of steel rod (1/4" to 3/8" in dia). You can use the old Marshalltown grooving tool....referred to as a "crack spoon" on my sites

. However, with uneven joints we tend to brush back the mortar about 7/16" with the wire brushes and then as it cures more you smooth off with your glove or soft brush. Some will even use a damp sponge for a really slick finish.
The recessed mortar finish works best with tighter laid blocks and it lets people see the stone for what it is...worts and chips an all. It all takes a lot of practice and is affected by the moisture variables of the atmosphere, stone and your masons sand. A steady hand and a good eye go into making your finish nice and uniform.
As a mason your finish is one of your trade marks. The more time you take the better you get and the more people are willing to pay for your work. Lots of people can give you tips and ideas but at the end of the day you have to teach yourself the skills.
I hope I'm not off topic here and this helps some.