They can take a pin, but they can also be soldered directly to a square or rectangular pad beneath them much as a SMD passive would. This particular form factor allows for both through-hole (with a bit of work) and surface-mount in the same package.
In answer to your questions,
Remove the third lead, this will let you clear/prepare the pads for re-soldering.
It's probably best to flick the old solder off the leads; do this by heating and bending the lead with the tip of the iron and letting it spring back, DO NOT DO THIS TOWARDS YOUR FACE!
You can buy solder with a flux core. Also, if you can get it, buy lead solder, it is MUCH nicer to work with.
As regards the vid:
Yes the guy tins his iron to remove the leads, you should always tin (add solder to your iron tip) before you do anything, it stops the tip oxidising. It also helps when melting old solder.
To clean the pads generally I would put a bit of new solder on, then wick it off to give a nice clean and shiny surface.
He is applying flux, this will stop the solder "balling up" and sticking to stuff it shouldn't. You can do this, but if you are careful, and have flux core solder, you wont need to.
As regards the rest of your questions, this is just about technique. The guy seems to tack solder (a connection made to hold it in place) the leads at first. When the whole lead is tack soldered he goes around and tidies up the job.
So what YOU need to do.
Remove the lead - Tin and clean the pads and flick the old solder off the end of the lead (NOT TOWARDS YOUR FACE!)
Add a little solder to each of the pads
Place each lead above the pad and push down gently whilst touching the tip of the iron to the solder you placed on the pad. Make sure you remove the iron before removing the pressure.
When all connections are made, get some tweezers and push down on each lead in turn, melting the solder on the pad with the tip of the iron so the lead is properly flat on the pad.
Remember! Before you touch the iron to anything, make sure you have applied some solder to it and wiped it off on a sponge.
Also, if you are buying new equipment, when you first turn the iron on, keep adding solder to the tip and wiping it off. You need to thoroughly tin/protect the tip before you start using it. You can get little pots of hard flux, this is useful and it will help remove the crud on the tip due to oxidation. Also, tin it and DON'T wipe it off when you are done, the tip will still be hot for a while after you switch it off.
Best Answer
This can be accomplished using soldering.
You basically create a set of pads on the rigid PCB which line up with a set of pads at the end of you flex PCB.
You can either join such that the pads extend right to the edge of the flex PCB and you solder with both sets of pads facing upwards. In this case the pins will be wired 1-to-1.
In this approach the pads are both tinned, flux is applied, then a soldering iron used to drag the solder up from the rigid PCB onto the pads of the flex PCB to make a join. This is not the strongest way to join the boards, so some additional flexible glue between the flex PCB and rigid PCB might be called for.
A second soldering option is to have pads on the bottom of the flex, not quite at the end of the board, and top of the rigid PCB. Then the pads are joined with solder sandwiched between the pads like a QFN style connection. In this case the pins on one board are mirror imaged as they will be bonded n-to-1.
For the second option, to solder, tin both sets of pads and once cooled add some extra flux. Alternatively apply solder paste. Then line the pads up, and heat the joints applying gentle pressure.