Handling mezzuzot and other Judaica from parents’ home?

I’m a Jewish person who wants to hear what people on all points on the observance spectrum think. But I’m not observant and don’t feel comfortable asking my own congregation’s rabbi for advice.

Anyhow, my elderly father has moved into a nursing home and is very sick. He’s not officially in a hospice program but probably would qualify to be in one. I live out of my town. My sister lives in town and is in charge of care.

She’s a loving, generous person, who has power of attorney, and who’s paying a huge amount of money for my father’s care, and she’s an extreme minimalist. She has a horror of hoarding and throws out anything she can throw out.

So, she’s already gotten rid of most things in my father’s home. She let me go through some specific things that I asked her to hold for me to look at. She has huge amounts of storage space in her house, but, from her point of view, me going through things and saving some of them (like: family photos, and old post cards and letters) is an act of insanity. She got really angry at me when I tried to save my father’s old suits, in case, somehow, he recovers and needs a suit. Her attitude is that, if my father recovers and needs a suit, that will be a wonderful miracle, and she’ll buy him new clothes.

Anyhow:

1. Are there any Jewish laws or traditions related to the handling of the personal belongings of an elderly person who’s very sick and near death? (What’s done is done, but, now, I’m thinking I should have done more research.)

2. I have two mezzuzot from my parents’ houses, and a box of matzoh covers, yarmulkes and tallises. I have theories about who wore which tallis but don’t really know. As far as I can tell, the matzoh covers and yarmulkes don’t have any version of the name of HaShem on them. I’ll probably keep all of that stuff.

But,

a. Are there any rules about how I handle matzoh covers, yarmulkes and tallises inherited from my parents?

b. Are there any different, possibly more strict, rules for handling mezzuzot? If, say, I keep them in a pretty case with the yarmulkes and the matzoh covers, would that be OK, or would that be bad?