There's kind of a lot to unpack in your post, so I will tackle some of it and try to get to some of the other stuff, but a lot will have to wait for someone with more experience than me for a firm answer.
First, the caps and the switches - there are several types of keycaps: MX-compatible, Topre-compatible, and a some other stuff that I am not too familiar with. The "MX-compatible" refers to the switch type - in this case a Cherry MX switch. If you look at the switches in the first link you posted, you can see the cherry logo embossed on the switch - this is the company that has manufactured the switch. The stem - the part poking out of the top of the switch - is in the shape of a cross. This shape determines which caps can fit on the switch itself. So, as long as the switches are "Cherry MX-compatible" they should fit the switches you have linked to.
The "SA profile" of the caps refers to the general shape of the keycap itself - take a look at this link for a good comparison:
https://geekhack.org/index.php?topic=83853.0In my understanding (this is one of the things that someone with a better understanding of things will hopefully have a better answer for), the SA profile is owned by Signature Plastics. So, the capset you are looking at is likely(?) a knock-off. This isn't necessarily a bad thing, just something I thought you might like to be aware of.
On to the soldering: soldering makes the electrical connections between the various parts of the keyboard and the PCB. And, unless the keyboard is pre-assembled, you will have to solder the switches to the keyboard. If you take a close look at the picture in the second like you posted, there are two small, offset holes in each of the spaces for the switches in the PCB (the square holes are the spaces for the switches). There are two "posts" (not sure of the correct term here) coming out of the bottom of the switch - these extend through the holes in the PCB, and are then soldered in place.
As for the soldering - that's something (I think) most of us have to learn. Take a look at this thread for a bit more info:
https://geekhack.org/index.php?topic=42824.0You can also Google around, it shouldn't be hard to find tutorial videos and such. You can even find keyboard-specific tutorials for soldering. You can jump in to soldering for a fairly reasonable price (you have to purchase a soldering iron and some other equipment), but if you are willing and able, there are lots of people on this forum willing and able to provide help and guidance along the way.
In your post, you mention Gateron switches - but you have linked to Cherry switches. Either
should fit the linked PCB, the caveat here is that some switches are made to be mounted directly to the PCB and other have been made to mount on a plate - between the switch and the PCB. You should check to be sure you have the right kit/switch combination.
As for the programming - I haven't done any programming yet. I do know that it will let you configure the layout to your personal specifications - so, if you want to change the position of the backspace key on the keyboard, or specify macro keys, this is how you would do it.
Again, there are some good resources in this forum, and a number of people willing to help you out here too.