Hi Sasha,
Axies do get startled when there is sudden light exposure, such as from taking it out from the fridge. This is normal and anticipated. You can help minimise this by covering the tub with a tea towel and then slowly sliding/lifting the towel away to uncover the axie gradually.
Axies also get startled when they feel they are being pursued when you try to catch or transfer it to another tub or even just the water turbulence during water changes. This is also normal and anticipated. You can minimise this by gently tipping water in and out of tubs and avoid jerking, abrupt, frantic movements when trying to transfer the axie.
You are doing nothing wrong on both accounts.
Due to the cold, the axie's digestion is slowed down. As such they tend to lose their appetites while being fridged. However, their metabolic demand and physical activity is also reduced and thus food requirement is also proportionately reduced. Axies can safely go without food for up to 2 months, and even for longer periods while being fridged. This depends on many factors such as the original health and body condition of the axie, the age of the axie and even the fridge temperature.
You can try offer live blackworms, chopped up live earthworms and pieces of fish and prawn while the axie is fridged. Try to offer smaller than usual portions of food. The smaller pieces allow for easier digestion. Live wriggly worms also tend to stimulate their appeitite. You can safely leave a small amount of live blackworms overnight in the axie tub until the next day during the water change. Any other food type, i would recommend you remove it after 30min if the axie doesn't consume it.
Based on the photos, i really don't think there was infection. I am more inclined to think it is normal skin shedding coupled with small physical trauma tears around the tail. Axie fins are very delicate and friable and very often they can get torn off. They do tend to regenerate uneventfully. There are many fine blood capillaries around the tail which are more apparent in light coloured axies. That is normal.
Cheers.