Yes you will you will, either on each LED or a board of sorts. These are surface mount LEDs so you can't wire them in free space.
Each of these LEDs require a surface mount ceramic 0.1uF capacitor across the power and ground as well as the 100uF at the start of the strip.
Here are some pictures of mounting them on a double strip of self adhesive copper, wrapped round a circular box.
This is the box and copper, the hole is so they can be shining into the inside of the box.
Well it is not the best and seems to be very expensive. I don't know the exchange rate on that page but these should be about $0.05 US at the most.
The problem is the long leads, they act as inductors and make them stop looking like a capacitor at higher frequencies. Something like this is better. Surface Mount
No because these capacitors are not polarised, and if they were then they would not work. The large 1000uF capacitor are polarised and they work as a team with the ceramic ones. You need one ceramic one per LED and only one large capacitor per strip, if the strip is longer than about 12 or so LEDs.
Either will do, a short strip will only need 100uF but a longer might need 1000uF. It surprises beginners that some electronics is not calculated to the Nth decimal place. You can't have too much but you may have too little. You haven't said how long a strip you are going to make.
Sorry I don't understand you. The small capacitors go between the positive and negative supply rails, the closer to the actual LED as you can get them.
Well again anything between 200R and 500R will do. I tend to use 220R (that is 220 ohms).
I actually use WS2812b's to stick them into small buttons that otherwise may not be illuminated.
Yes you can, but keep the leads as short as possible. For ceramic disk or poly-propylene, polarity is not an issues. For larger capacitors like 100uF or 1000uF that usually come as electrolytic capacitors, polarity is relevant, and if you connect them the wrong way round, they may explode (mind your eyes)