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.
It sounds as if one of your problems is the high temperature needed to melt the solder. There is a commercial solution called "Chip Quik" ChipQuik that should help.
Basically, this is a very-low melt point solder that is added to the existing solder on the board. The resulting mix of solder metals has a much lower melt point than regular solder and you can usually manipulate the plastic bobbin without melting the plastic while the solder is still molten.
You have to be careful to remove all of the contaminated solder before resoldering the both the removed part as well as the PC board but that is easy with any of the standard solder-removal techniques (solder wick, vacuum desoldering station, etc).
ChipQuik used to send out free samples, enough to remove a few chips. The company was recently sold and I don't know if they still do that - but it's worth asking.
The free sample I received many years ago convinced me that the product was effective and I now keep it around on both my lab bench as well as our rework stations.
Best Answer
I also would usually solder a test pin or wire on there, but if I happen to be using a part with legs instead of a qfn or bga, I sometimes reach for the micro grabbers. These are small enough to claw onto single pins without shorting the adjacent ones