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.
If possible, I would recommend removing the damaged through-hole transistor by cutting its leads above the board surface, then simply trim the leads of the new one to where you can install it by simply placing them on the existing solder pools & melting the existing solder (possibly with adding a very slight amount).
This may not be the "prettiest" way to effect the repair, but since you seem to not be very experienced with soldering on PCBs with SMD devices (and don't necessarily have great SMD soldering equipment), it's probably very much the safest.
Best Answer
A good DIY method for the occasional "home reflow solderer" is the electric frying pan technique. It allows you to heat the MCPCB very evenly. Almost any $19,99 model from a thrift shop will do the trick.
First, clean the MCPCB with Isopropyl alcohol. Apply a bit of flux with a flux pen. Then squirt a minute bead of solder paste, use one that already comes syringed, on the anode, cathode and heat-sink contacts of the MCPCB. Next place your LED in the right location. Turn on the electric frying plan until the solder paste liquifies. You will see that the surface tension will suck your LED in the right location.
With this technique, you can even "home-reflow" intimidating looking DFN ICs and the like surprisingly accurately. Sparkfun et al. have good examples of this in their old video library here.