Welcome to the forum!
I've tried answering your questions here:
1. That pump looks like it is probably intended for filling soccer balls, basketballs, etc., no, it won't work as a bike pump. You will need a bike tire-specific pump. Ideally, you should have 2; one for pumping the tires at home (called a "floor pump"😉 and one to take with you in case you flat and need to pump up in the middle of a ride - a "frame pump", sometimes called a mini pump. etc. You can get away with just having a frame pump, but be aware that it will take longer than using a floor pump.
2. Almost all road tubes today come with Presta valves. Floor pumps generally have both Presta and Schrader (a fatter valve type usually seen on cheaper department store bikes).
3. No, it won't leak, as it's held in by air pressure, but if you accidentally touch it, it will let out air, so it's best to screw it shut after pumping. You don't need the black plastic cap, in fact, just throw it away.
4. A seat pack goes under your saddle to store stuff you might need on your ride, ie, patch kit, tire levers, money, tools, and yes, sometimes a mini-pump (if it will fit, depends on size).
*Some people just put their mini-pump in their jersey pocket. Other people (myself included) use air cartridges instead of a pump, as they fit easily in a saddle bag and fill the tyre very quickly. However, they are a bit pricy, and can leave you stranded if you have multiple flats. YMMV.
5. The tube holds the air and goes inside the tyre (what is connected to the rim and contacts the road).
You might want to get a book that describes the basics of cycling and/or maintenance, Park Tools have some excellent ones, they also have a lot of info on their website.
Happy riding!