I've seen a few cacti change the color of their skin but the skin doesn't become squishy. I'm not sure why they do that. I have a cactus with its whole lower half a darker shade. It looks like rot but the cactus has been like that for a few years, the lower part hasn't become mushy, and the upper part still looks healthy. I'm guessing the color change can be a sign of rot and other times isn't. I don't see the point along the stem the roots start on the cactus. Looks like the cactus might've been planted a little too deep in the soil. That can invite rot. Cacti can poke each other and leave spot scars but the brown spots in your pictures look different. I'm not concerned about the spots in the lower picture. They look like they aren't spreading. However, the spot in the lower left quadrant of the top picture has a trail of lighter skin running down to the lower area of the cactus. Is the lower part of the cactus unusually soft and moist? If not then I'd clear the soil away from the lower part until you find the point the roots start coming from the stem and put that point level with the top of the soil in the new pot. Then watch for changes of the spots and lower skin for the next few weeks. Or, if the lower part seems rotten you can cut it off by cutting above the rotten area in an area that looks healthy. Let the cutting callous for some time and then plant it in a pot. However, cactus cuttings can be tricky. Or, at least, for me they are. If you don't have experience with making cuttings and rooting them then research it.

Mealy bugs? Can you see oval critters that move when you touch them? My first thought is to put in the shower and gently blast them off. I don't know if any of the plant masters would agree with me. I hate using chemicals.

It seems to me that something has been munching on the leaves... Putting succulents outside ..."maybe once a week or so to get extra sun"... isn't very good IMO. They need lots of good light constantly, and should be acclimatized to outdoor sunlight. If you take them in, it is putting them into lower light again and again. I suspect that plant is growing somehow etiolated - the second photo is showing stem being quite light in color and leaves further apart. Is there a flower stalk (3rd photo)?

My Agaves are all outside. I get a lot of rain - 1.5 to 2.0 meters - during the summer. The reasons that my plants can survive is 1.) I use a gritty mix that drains very well and 2.) my rains come during the hot weather, not the colder winters. I don't know what materials are available to you in the UK, but others on this forum may have some suggestions. Because of my summer wetness, I use a mix that looks like this - no potting soil or organic matter. Maybe you can move your plant outdoors during the warm months? I can't tell how large or heavy your plant may be and would hate for you to have to buy a trolley! LOL But even overcast or diffuse sunshine is better than light coming through a window - especially our new energy efficient windows. That's just a couple of suggestions based on MY experience. Yours may be different, of course. I agree that your plant will bounce back. Nice looking Agave!