I think your pot is too big and too deep. It should only go into a pot one size bigger. They do not grow very fast in pots. you need less than one inch all around for space, otherwise you run the risk of the media being too wet.
I think it is one of the coryphanta since the spines are on turbucules and not on ridges. The general used to be called mammilaria.
I use newspaper or tongs with a wash cloth or piece of sponge tied with a rubber band on the tips. The tongs are good if the plant is not that big or heavy. For the larger plants, leather gloves and the sports section of the newspaper folded length wise until it is about 2 -3 inches wide. you would use it like a strap around the cactus. It is soft enough not to damage the cactus and strong enough to pick up a fairly heavy cactus. You may still get stuck, but if you are careful and especially if you can get a second person with another strap on it it helps.
Here is a link for different catus mix recipes
For myself since I tend to over water and my succulents are outside in the rain. I keep them away from my sprinkler system so they have to be hand watered. I use the smallest and shallowest pots I can fit them in. I prefer cement and clay because cacti are top heavy so they make a better counter weight than plastic pots. Clay and cement pots also breathe so they dry out faster. For the bigger and taller plants I do use plastic pots but I make sure they have a lot of holes. I use number 3 rocks ( I get that at a building supply) and cinder ( I prefer large or medium, not crushed) Sometimes The cinder is still too fine so I add large perlite to the cinder. And I feed it with a slow release fertilizer as rocks do not supply much NPK. No compost and definitely no peat moss as they both would hold too much water. If it rains hard every day, I have to bring them inside my patio or I may lose them.
Happy gardening in Hawaii. Gardens are where people grow.