Just go ahead and wire it to the small connector. The pin spacing is wide enough you shouldn't have any issues with bridging or shorts. Pre-tin the wire and the connector, and add on each one. You could strip each wire a bit longer and twist it around the connector pin if you feel it is not strong enough.
A small piece of heat shrink is all you need to cover up the small connector afterwards. If you are really worried about shorts, you can add heatshrink to each individual wire before soldering it on, then still add the bigger heatshrink afterwards.
An alternative way to attach it would be by using a small piece of protoboard. Solder the connector to the board, then use wires from the protoboard to the main board.
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
Those tabs are designed to have wires soldered directly to them. Trim the insulation so that it comes to the end of the board and the wire overlaps the entire pad. Pre-tin the pad and the wire and just melt the wire into place.
The two circular holes at the one end are for the XT60 connector which you will find on many LiPo batteries.