Around the contact pads I've soldered the board (and on the pad and joint itself, actually) has some yellowish stuff on it (please see photo attached), at first I thought it was the board itself melting slightly or delaminating, but is it flux residue?
Yes, it is flux residue. Since you are using no-clean flux then you can just leave it there. You can remove it if you want with isopropyl alcohol (70% is OK, 99% is better, don't breathe it in) or with commercial (expensive) flux remover. It's usually not necessary to remove it unless you are concerned about appearance or long-term reliability. Flux will absorb moisture over time and sensitive electronic circuits will see an additional leakage path through the flux due to this moisture. But I doubt this will affect what you are doing.
Also, the tip of my iron was tinned when I got it (i.e. the last 5mm of the tip was silver whereas the rest was duller) but after a little bit of soldering (and loads of cleaning, I dabbed it on the sponge after basically every joint made because of the build up of brown/black stuff on it) the silver area is a lot smaller.
Don't use a sponge. Use brass shavings:http://www.hakkousa.com/detail.asp?PID=2989&Page=1
While you're there browse through their great application notes on proper soldering temperatures, tip maintenance, etc.http://www.hakkousa.com/doc_library.asp?DocType=Tech%20Note
The brass will keep your tip a lot cleaner without cooling it, thus allowing you to keep the tip on a lower temperature setting and preventing oxidation.
Am I supposed to just keep applying solder to it to tin it (even though when I did that it just stuck to the bits that are already tinned and turned into a big drop of solder)?
Yes, you must always keep solder on the tip. It is the first line of defense against tip oxidation.
Finally, is 370C/700F a good temperature for soldering or would 310C/590F be more suitable?
Turn the temperature down, increase the size of the tip if necessary. 590F should be doable (I usually use 650F but should probably turn it down) but 700F is definitely too hot. If your joints aren't heating up and solder is not flowing easily after 1-2 seconds, the tip is not big enough. You not only need temperature you need thermal mass to transfer that temperature to the joint. You absolutely positively definitely no-questions-asked need to get an assortment of tips of various sizes, then absolutely positively definitely CHANGE the tip to match the joint to be soldered. Plan your work, solder the small stuff first with a small tip, let the tip cool, change to a bigger tip, solder bigger stuff, etc.
And finally, I really doubt your soldering iron has enough thermal mass to delaminate your board.
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