Way OT but let's do this. Note: climb V5/6 currently.
1. Yes that is bouldering, but bouldering is a subset of rock climbing, as is sport climbing.
2. Bouldering is frequently about balance and power, for some problems flexibility isn't as crucial. I climb in jeans outdoors, sometimes it's the best option as rock can quickly cut through lighter materials. My jeans are skinny and stretchy.
3. Climbing works your core, simply by climbing more and more.
4. Don't play the grades game, don't buy a hangboard. Go into a gym and have fun on the boulder. Don't try too hard and hurt your tendons, they take years to grow accustom to the stress induced by hard bouldering.
SF is very fascinated with Yosemite, but there are other areas that also push the sport. Obviously, the Europeans have done alot. Bouldering is itself a pretty hardcore activity, and one that has pioneered at times certain techniques that have had broader impact. Some of the best in also in the USA: http://en.wikipedia.org/wiki/Hueco_Tanks
Where are you based? I picked up climbing in May 2012, and can offer you some advice based on what I've been through. Email's in my profile to continue the chat if you want.
Find your closest climbing gym and show up. I have very limited climbing experience, and only in gyms, but it's one of those things that it's just easier to show up.
OT(as far as the OP is concerned) questions from someone who is considering taking up climbing.
1. Isn't what you are doing in the video called bouldering?
2. Do you usually climb in your jeans? I am wondering does it allow you required flexibility?
3. Do you work on your core for climbing or climbing works your core?
4. What would you say about artificial walls and bouldering arenas for a beginner?