First, take off all the doors, drawers and counter tops. Many times, counter tops are glued to your base cabinets, and are bolted together at the seams with pocket bolts. Try to take these off carefully if you plan on re-using them, or donating them too.
Then, look for a board near the top of the base cabinets, this is the board usually used to screw the cabinets to the wall. Take the screws out that are being used to secure them to the wall, and it's likely that they'll can just be pulled away from the wall. Be careful though, sometimes, the base cabinets are screwed to each other too, so look at the insides of the base cabinets to see.
The same basic process can be done for the upper cabinets as well, except that you should make sure that they are not screwed to each other before trying to take one down. This is almost always a two-man job.
As for donations, I suggest looking for a Re-Store in your area. They are operated by Habitat For Humanity, and I personally have purchased many things (including cabinets) from them. All the proceeds from the sales of donated items goes to help low-income families earn the pride of home ownership!
I have built many islands using top cabinets - they don't have bases. The easiest way to add a base is with some 2x4s (or 2x6s for height). You will just build a simple rectangle with bracing for each cabinet. I guess you could build a big rectangle for a row of cabinets but I have been down that road and it won't be traveled again.
Basically you just need to decide how you want this to look. Do you want a ledge (common in america) or do you want it flat? Ahh and then the last thing. You will need to put on a face plate on this section that matches the rest of your cabinets. If you are going to have a 4 inch ledge... no big deal just make sure all of your rectangles project at the same depth. But if you want it flat then you really have to be precise and take account for the depth of your face plate.
About Brick. I wouldn't even think about it. Brick gets its strength from being surrounded by other brick - kind of like tiles. I don't see it being strong enough on its own to handle the weight, movement, or whatever unless you build a solid brick brick. Also you are going to deal with a ton of other issues like how do you secure the brick to cabinet and how do you keep the bricks even across all cabinets... Can brick be done? For sure. But expect a ton of time and it will need a few rows for sure - and I am not sure what you plans are if it isn't close to level.
Another alternative is get some screw in cabinet legs... we buy them at IKEA here.
Best Answer
Actually, I noticed two problems with your countertop. One, the gap does seem too large for a stone countertop. Two, the seam is perpendicular when it should have been cut and fitted at a 45 degree angle. The edge, or bullnose would fit together much better if cut and fitted at 45 degrees. I suspect this was stock material installed by someone not equipped to do it properly. Normally, the joints would be precision fitted at a 45 degree angle at the stoneworks before installation. The joint on a quality countertop would be exact, epoxied and polished. In your case, there is no real, easy fix. The gap would have to be filled and sanded. Then the surface would have to be polished to bring back the original finish.