Sadly, this soil is not great for succulents. I don't grow cacti, but my succulents all had problems in this soil. Kind of makes you mad when you realize that you really didn't find the perfect soil after all, huh? You need to add at least 50-60% perlite (or some other gritty substance) to give adequate drainage and "wiggle room" for the roots. My plants seemed fine for about a year, then the soil was compacted around the roots and not even absorbing water any longer. Your plant will appreciate a grittier mix. You can search on this forum for soil mixes, gritty mixes. But for the immediate remedy I recommend adding the perlite to your soil. Regular sand is not recommended, but some use the coarse builder's sand for grit.

Most likely fungus gnats which are attracted to the potting soil of indoor plants. Control with BT. extensive information here. Also repot your Cactus in a gritty mix more appropriate for xeric plants. What growers use is junk; they can get away with it because they grow under precise, controlled conditions. Use tongs to hold the plant and wear gloves. Do not touch the spines with your bare hands. (You undoubtedly know this, but it bears repeating over and over and over...)

Nothing wrong with it. You have the variegated and monstrose form of Opuntia monacantha. It may have been in the dark for a long time, and so lost its variegation. You probably put it in better light, and now you can see the white again.

That happens when they don't get enough light. Give it more light. Gradually increase the light so you don't burn the cactus. The new growth on the cactus in the lower right corner might be a bit thin, too.