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.
There are two values you need to worry about: voltage drop, and power dissipation. Both are simple Ohm's Law and are functions of the trace resistance.
The trace resistance is a product of its cross sectional area, and its length.
Reduce the length and you reduce the resistance. Reduce the width and you increase the resistance.
So you can have a shorter run of a narrower trace and still handle the current.
The formula for calculating the resistance of a trace is:
$$
R = \rho \frac{l}{A} \cdot (1 + (\alpha \cdot \Delta T))
$$
- \$\rho\$ is the resistivity, which for copper is \$1.68×10^{-8} \Omega/m\$.
- A is the cross-sectional area in m²
- l is the trace length in m
- \$\alpha\$ is the temperature coefficient, which for copper is 0.003862 at 20°C.
- \$\Delta T\$ is the temperature difference from 20°C
So for a 300 thou (7.62mm) trace at 1oz, which is a thickness of 0.0347mm, a rectangular cross-section would be
$$
0.00762 \times 0.0000347 = 0.000000264m²
$$
Of course, with etching and other factors it won't be as thick, nor perfectly rectangular, so reduce that a little - let's say for the sake of convenience it's 0.0000002m².
Then you have a trace that's 0.05m long (5cm). What is the resistance of that trace at, say 23°C?
$$
R = 1.68×10^{-8}\frac{0.05}{0.0000002} \cdot (1 + (0.003862 \times 3))
$$
$$
R = 1.68×10^{-8} \times 250000 \times 1.011586
$$
$$
R = 0.00425\Omega
$$
So once you have the resistance, and you know the current, you can apply simple Ohm's Law to it. Say 15A, your upper value.
The voltage dropped across that trace is
$$
V=IR = 15 \times 0.00425 = 0.064V
$$
The power dissipation will be
$$
P=I^2R = 15 \times 15 \times 0.00425 = 0.956W
$$
So now you can calculate what the voltage drop and power dissipation would be over your small traces to see if it's tolerable.
There are also various tricks you can employ for handling larger currents. One of the most common (and old-school) is to leave the traces unmasked, then flood them with extra solder. This massively increases the cross-sectional area thus reducing the resistance. You can also use electro-plating to achieve a similar result, though this is considerably harder to do, especially in just a small area of the board.
Using wires instead of (or as well as) traces can also be done.
As an aside, you should also consider if the connections, and the pins used in your connectors, are suitable for carrying up to 15A.
Best Answer
No reason you can't use a standard header pin, or a wire wrap pin, wrap it around then solder.
Alternatively, a copper landing/square/point can be used.
Often small gauge wire is soldered to test points for debugging or hacking purposes, and that's essentially what you'd be doing.