I think you're going to have a lot of problems trying to prevent water/moisture in this garage.
As you say, improving the gutters is a good first step. It's likely that there is no drainage system around the foundation of this garage, so that is going to cause some trouble. You may be able to mitigate some of the ground water by putting a french drain around the outside, and draining it off to the side somewhere away from the garage. It doesn't need to be too deep (6" is probably enough), but there should be a gravel bed wide enough that any of the water that would drain around the walls is diverted (remember, water will follow the path of least resistance).
Ultimately, despite drainage, you're going to have problems with condensation. As the warm humid air meets the cool foundation, it's going to form condensation on the walls. This is unavoidable, since it is a garage and has a giant opening to the outside. If you build the frame around the inside in such a way that there is still air flow for drying, you may be able to reduce the effect though and at least somewhat prevent the smell. Definitely use pressure treated wood for everything here.
The "traditional" way to do it is, as you have suggested, to cut the bricks out by hand (and refit the cut bricks once you've done to give a tidy edge). It can be a time consuming job.
The more usual way to do it once the lintels are in place would be to use a disc cutter - usually a petrol driven one - (aka a rotary cutter or Stihl saw, after the common brand) with a diamond-tipped blade to cut through the bricks. This will make a lot of dust. You can get a water pack to reduce the dust (and cool the disc, reducing the wear on it), but it can still make for a messy operation, as the water will make the dust into a slurry that the disc will spray about.
You can hire the saws from pretty much any plant hire outlet, who will also hire the diamond tipped blades, charging you per milimetre (or half milimetre) of wear. That can be expensive though, so if you've a lot to do, it might be cheaper to buy the diamond-tipped disc from an independant supplier.
I've also seen what is basically a chainsaw with a diamond tipped chain for cutting through walls, etc, but I've not actually seen one used to comment as to how effective they are.
Don't forget that you should use eye, ear, and dust protection if you're doing work like this.
Best Answer
Several things are occurring here: 1) cracks through the blocks not just mortar joints, 2) vertical and diagonal cracks
1) In construction, block walls are designed with mortar joints to be weaker than concrete blocks. That way, when the building settles (and all buildings settle) the wall will zig zag crack down the wall in the mortar joint, NOT through a block. Replacing a block is much harder that repointing the mortar.
Your mason has screwed up and caused a big problem that can only be fixed by replacing the cracked blocks.
2) Generally, vertical cracks are caused by the foundation settling in one location and a diagonal crack is caused by the foundation settling along many feet (big area).
They are both “structural” cracks, it’s just that one is a bigger deal than another. A small crack (like yours) is called a “hair-line” crack and is minor. Often the building is constructed so fast that the foundation does not have a chance to settle before the final finishes have been installed. I think this is what happened to you.
I’d monitor it by drawing a line across the crack every 4’ or so, number the lines, and measure and photo each crack. I’d keep a journal for 1 year checking at regular intervals, say on the 1st of each month. If it continues to get bigger, you have a problem.
Likewise, if any cracks develop, note and monitor it too. If crack gets to be more than 3/16”, you have s problem. Let your contractor know what you’re going to do and send him notes and pics each month. Let him know you expect the cracked blocks replaced at the end of 1 year, plus and walls with cracks larger than 3/16”. (Check your warranty and make sure you send certified letter before the warranty ends. Maybe this will be a 10 month monitoring process instead of 1 year.)