Reddit mentions of MG Chemicals #3 No Clean Super Wick Desoldering Braid, 0.075" Width x 5' Length, Green

The soldering iron you have is fine. I bought my batteries from console5.com . As far as other things, I would suggest some flux and a desolder braid/wick. If you don't have solder I would grab some of that too. I've got alphafry solder, mg chemicals desolder wick, and sra flux.

Batteries:
https://console5.com/store/panasonic-cr2032-3v-lithium-battery-with-solder-tabs-pins.html

Desolder wick:
https://www.amazon.com/gp/aw/d/B00424S2C8/ref=mp_s_a_1_6?ie=UTF8&qid=1525828012&sr=8-6&pi=AC_SX236_SY340_FMwebp_QL65&keywords=solder+wick

Flux:
https://www.amazon.com/gp/aw/d/B008OC0E5M/ref=mp_s_a_1_3?ie=UTF8&qid=1525828121&sr=8-3&pi=AC_SX236_SY340_FMwebp_QL65&keywords=solder+flux+pen&dpPl=1&dpID=31u2-ijUmFL&ref=plSrch

Solder:
https://www.amazon.com/gp/aw/d/B00030AP48/ref=mp_s_a_1_3?ie=UTF8&qid=1525828185&sr=8-3&pi=AC_SX236_SY340_QL65&keywords=solder

You certainly can use that soldering gun but it won't be fun or easy because it is so large (that's what I started with and used for a few years). I'd suggest picking up a Hakko FX-888D or Hakko FX-951 if you have a desire to continue with electronics.

Definitely add more solder because it has flux in it which will clean the oxides and allow the solder to flow out. You will also need to use a bit of solder wick/desoldering braid to clean up the pad before fixing it. You'll want to get some Kester leaded solder to work with.

Watch these Pace soldering tutorials before you begin to understand the basics. Heat the pad and wire then touch the solder wire to the joint. If you add it to the tip of the gun/iron the flux will burn away and it will start to oxidize. Doing it that way certainly has it's place but it isn't for this type of work as you'll need to add extra flux to protect the joint.

Use a solder sucker or some desoldering wick to remove the excess solder that is touching the logo.

Something like this https://www.amazon.com/gp/product/B00424S2C8

Could be a few things but mostly likely the wrong wick for the job. If you get wick from the hardware store it won't work well. That wick is designed usually for plumbing size jobs and higher temps.

For PCB size jobs and SMD components getting the right wick is important just like using the right solder. Something like this may do well for you:

MG Chemicals #3 No Clean Super Wick Desoldering Braid, 0.075" Width x 5' Length, Green https://www.amazon.com/dp/B00424S2C8/ref=cm_sw_r_other_apa_Fe5Dzb8XT56FK


I also use this wick and like it:

Aven 17542 Desoldering Wick, 2.5mm Width, 5' Length https://www.amazon.com/dp/B003E48ERU/ref=cm_sw_r_other_apa_9g5Dzb43FFNZ3



As far as technique, clean then tin your tip, heat the solder you are removing, re-tin your tip quickly and heavily, apply the braid and your tip on top. Solder should flow into the braid very quickly now that the solder was pre heated on the bottom of the wick and the tinned iron tip on top heats and flows into the braid sucking up the solder from below the wick.

This helped me. It helps pull out the solder I was able to just remove all the solder holding in pins with it. MG Chemicals 400-NS Series #3 No Clean Super Wick Desoldering Braid, 0.075" Width x 5' Length, Green https://www.amazon.com/dp/B00424S2C8/ref=cm_sw_r_other_awd_xK6xwbDCA2VBF

You might find a solder sucker - https://www.amazon.com/WEmake-WM-SP4-Solder-Sucker-desoldering/dp/B0002KRAAG.
Or some de-soldering braid/wick - https://www.amazon.com/MG-Chemicals-Clean-Desoldering-Length/dp/B00424S2C8 useful for removing the solder easily when removing header pins. Also before you try de-soldering, remove the plastic part from the pins by pulling it off with some pliars, this allows you to remove the pins 1 by 1, preventing the solder pads getting to hot and coming off.

Get solder wick (not the exact one I used, just one of the first ones I found on Amazon) and make sure to get your soldering iron hot enough. I first tried at 300 C and was having a heck of a time. Tried again at 400 C and everything went much smoother.