High precision soldering of through hole components
That is not tiny, and not even close to close (pun intended).

I'd say the spacing between any two leads is abou 1mm, so you can use your soldering iron if you can buy a thinner tip. Thin solder can help a lot too, a bit of flux would be a great help but a waste for this kind of work in my opinion.

With a steady hand and less than 10$ of additional equipment (tip+solder) you can make a job that will look (and be) much, much better than it is now.

Desoldering the connector will be much more trouble. These are 13 pins, you will need to remove some of the solder using a copper desoldering wick braid cable or a small desoldering suction spring pump, then carefully pull out the component as evenly as possible, probably pulling out one side of some half mm, then the other side of a bit more, and so on.

As suggested that looks hand soldered because of the bent tips, through hole components are normally soldered with wave soldering anyway.