The washing part is obvious, but I am a poor one to tell you, since I don't do it often enough, myself. Collapsible canes that are held together by an internal elastic cord need watching. Keep track of the wear on the elastic cord, and be smart enough to replace it the day before it breaks. Of course, it is better to be a month early than a day late.
Cane tips last me anywhere from a few weeks to a few months. I carry a spare tip with me most of the time. A cane tip with a hole in it sounds different from a tip without the hole, and that is the sign to carry a spare tip all the time. I have worn out or lost tips unexpectedly. The unprotected end of any cane, especially fiberglass, is damaged quickly when rubbed against concrete. Just wave the cane, and keep the tapping to a minimum.
Does your cane have reflective tape on it? If not, you could put some on it anywhere along the stem. It is an investment in night-time safety. Reflective surfaces need to be kept clean or replaced to maintain their reflective value.
If you associate with other blind people, as I do, you may want some unique mark on your cane. I write my name in braille on Dymo tape and stick it on the bottom end of the handle.
When answering this question, you discover the great advantage of the folding or collapsible cane. Those styles can fit in a pocket, purse, on a lap, or under a chair very easily.
There are two horizontal dimensions and one vertical dimension. Find some place out of the way; lying on the floor under a chair or table, standing in a corner, or leaning against a wall. Be sure that the cane is lying flat on the floor and not resting on something that holds it an inch or two above the floor where it will be just high enough to trip the unsuspecting passer-by. In some crowded areas "up" is the only way left. When I am seated, I sometimes lean the cane from the floor to my shoulder, hooked behind my heel.
Once in a restaurant, I lost the tip while retrieving the cane from a tiny place behind the booth. I remember that incident, and sometimes I take the tip off before jamming the cane into tight places. In air travel stick the cane in some out-of-the-way place, but do not let the crew take it away from you. The regulations are now on our side.
The most obvious answer to this question is that you cane with your dominant hand. I am right-handed, but I trade off when I carry a heavy object. There may be a landmark I want to check on the other side. When I am walking with someone else, holding on or not, it may be better to have the cane on the other side to stay away from feet or another cane. If someone is holding my cane arm, it restricts the movement. I don't want that. I have had enough practice with my left hand so that I am fully adequate, but I am still more comfortable with the cane in the right hand. The question of which hand you use is a matter of the convenience of the moment.