Interesting topic!
I'll answer the second question first.
I want to know people care about me on a day to day basis, and I want to know people care about me in exceptional or more trying circumstances. To be honest I'm really quite selfish. While I can appreciate the effort people may expand to try to care about me, I really want them to do it in a way that feels meaningful to me. I indulge my emotions in that way, I guess...
I really appreciate having others present, checking in, and simply being a part of my life. Then there is being willing and able to listen and understand what I am communicating or why I may be feeling what I feel. I can explain the whys for them, if it's not evident to them, but I need them to be on the same wavelength as I (whatever that really means) and able to really grasp what I'm saying. I also appreciate people who are positive, solution-focused, and strength-oriented. If I'm in dodo, I want someone who can be a soft spot if I need to rest, but I also want someone who can do more than give me reason to believe I should feel hopeless and helpless.
I suppose I care for others in some of the ways in which I would appreciate being cared for. I make sure I am present. I listen. I check in. I try to be friendly and express caring on a regular basis so if shit hits the fan they know without a doubt I will be there. I try to listen for what the person is saying and what they may need, I try to keep an open-mind, and I try to consider the story from the other party's perspective. I also strive to check my own biases at the door. In general, I tend to be someone who likes to remind others of their strengths, and while the words I most frequently express are oftentimes "I understand", and while I'm inclined to let someone vent or spill their guts because I believe in the value of that, I expect on some level that they will eventually want to pick themselves up or will ask for help in doing so. I don't mind helping someone learn how to do something they don't know how to do, but I don't welcome/appreciate complaining or victimization. It's tiring and there are better things to do with our time. I definitely have my flaws when it comes to caregiving and caretaking. I can be impatient and obstinate, and in some ways too rational rather than feeling oriented, but I have some strengths as well and I think I express them pretty well on a day to day basis.