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's easy if you can measure the signal levels (Vdc, Vac) and know which direction from /to and cut tracks. The Sig Gnd should be directly oppsite the signal. Verify with Ohm on DMM.
Your mobile audio levels will be ~1V while many signals here could be <100mV so some resistor dividers may be needed. e.g. 1k:100R if there is no DC .
But where there are inputs , they are connected to outputs and you don't want to be driving them with a headphone jack signal when selected. If you can identify the cassette port and cut the track to the source with a sharp Exacto knife (with care) then you can use the sGND for signal ground close to L/R. It May be 1V from a pre-amp or not , test it if possible with audio. (get a good DMM >30$) or a cheapo <10 with good probes.
I use twisted pair fine wire to go to a 3.5mm jack such as AWG30 single strand magnet wire , pre-tinned at both ends by burning thru the insulation with solder then using a tiny needle-nose ( get some good tools) to make a hook or 1 turn around an exposed pin. Apply a tiny amount fresh solder and make it look neat, while holding it very steady for a few seconds.
I also use Kapton tape ( see pix) ( from Bangood) or polyurethane adhesive (Home Depot)for big parts like audio jacks, which takes a day to harden but if applied liberally will be stronger than a laptop jack. You can also use instant adhesive to tack in place with pressure and squeeze 10~ 30 seconds. But it may break away easily but hold long enough for the architectural subfloor adhesive to secure it to the board or box. Leave for 1 day or 2 when done to harden.
Here is what rework wire looks like with orange Kapton (polyamide) tape ( wire looks like AWG28? used for wirewrap) But I use magnet wire then make a few twists per cm, check continuity with DMM.
For finding high impedance (Z) inputs sometimes I'll power it up and run my finger over the suspect pin and check for hum, then I know it is an high Z input. Using a DMM to measure signal is Vac is also useful when operating on target pins.
Best Answer
I'm a pretty new electrical engineer, but I do a decent amount of soldering at work (I work as a test engineer and I have to layout PCBs for testing a myriad of electronics).
Depending on the frequency of the trace, lifting a pad can mean a new board for me. But if these are just test points (which I agree I think they are) and you don't know where they lead to, then you couldn't get any use out of them anyway... so I wouldn't sweat it.
As far as fixing lifted pads, I think the best bet is to try and flow solder into the pad area. Make sure the solder is also connected to the incoming trace. You should scratch some of the soldermask away from the trace to ensure a good connection. It's not nearly as nice, and you will burn the dielectric doing it. And if its a trace carrying a +100MHz signal I would be very wary.