I also have a Hakko desoldering gun pretty much the same as yours. What I found is that it operates quite well but maintenance is almost constant. Replacement of thin white filters, not thicker ones, cleaning of the sucker tube and in my case the replacement of the rubber disc in the vacuum chamber made it worth owning. It doesn't do its job well if any of the above are not done.
I normally just use my soldering iron (Hakko 936) with about a 3.2mm chisel tip and a good solder sucker (Edsyn, there are other good ones). I don't own a desoldering iron, and haven't really had the need for one. The larger tip is key though. Using the stock 1.6mm tip is much more difficult.
Besides flux, their low-temperature solders meant for desoldering contain tin, silver, bismuth, indium and copper, according to their MSDS sheets. They have a leaded variant that also contains lead (by far the most dangerous part) and antimony. I haven't seen anything about cadmium, fortunately.
https://www.chipquik.com/store/index.php?cPath=200
