I have not done the entire PCT, but I have extensive experience in the Sierras and on parts of the John Muir Trail. Do not worry about bears, consider yourself lucky if you see one, same for lions, just try to look big, don't bend down to pick up rocks as they can see how puny you really are and be mellow, they can smell your fear. You do not need a gun, yer safer there than in any city, but a hiking stick can be reassuring.
You don't camp in campsites if you can avoid it, the bears will travel a route and hit every campsite and not even come around if you camp off trail.
If you do need ice tools and don't have them you could be in deep dodo especially if you leave early, however the snow isn't off the high country until late July or August. Maybe leave the axe at home, but carry some cleat thingies to put on your boots.
Footgear - you didn't mention this. If you stay on trails and you don't have a heavy pack, you might be ok with trail runners if you have good solid feet with no problems and strong ankles, BUT the trail can be very rugged and washed out and even on the trail it can be a bad place for inadequate footgear. there is a point where the weight of medium leather boots will cause less fatigue and provide enough more security of footing to be easily worth their weight. If it was me, I would have real ankle high medium weight hiking boots AND trail runners for easy trails and stream crossing.
Jim YMMV good luck
If you do get off trail and who wants to stay on the trail in beautiful country like that, then be sure to have boots with vibram soles and gaiters are a must.
These are my own opinions based on wisdom earned through many wrong decisions. Your mileage may vary.