It sounds like either the shoes are not the right shape or fit, or there is a pressure point on your big toe. Or maybe you put too much weight on your big toe rather than spreading the weight across all of your toes?
Usually yes, but it should really be more of an uncomfortable feeling rather than actual pain. If you are used to climbing shoes, it may be fine, if not, you get used to it pretty quickly, assuming you didnt size down too much.
As long as you feel that they are the right fit for you, then yes. I remember when I bought my first real pair of climbing shoes after my beginner shoes, I tried downsizing as everyone had told me I have to, but even the salesman asked me to not downsize because the it just didn't fit me properly. What ended up being a "performance" fit for me was actually my street shoe size. This is in spite of everyone telling me I have to downsize to the smallest size that I can stick my feet in, regardless of the pain. So as long as you don't listen to that advice and you get what ever you can climb in without terrible pain, or without any pain at all, it should be fine.
Slightly, but not too much! Though, many climbers may say its just a matter of how comfortable you are with curled toes. If its not painful and it doesnt negatively impact performance, maybe its ok.
Honestly, getting the right size and just wearing them will really get the job done.
If your shoes are just not the right size though, then it maybe either not get better, or it may be a very long process. Not being the right size also includes shoes that are too big, so keep that in mind.
One thing I like to do is to bend then around and scrunch them up before putting them on. I do this for about a minute and it really helps soften then up and makes them much less painful.
There isn't really a "supposed to" or not when it comes to socks. You either prefer them or you don't.
Some wear them due to hygiene, some due to shoe sizing, and for some it's just comfortable.
Undoubtedly though, the majority of climbers do not wear socks with climbing shoes.
Generally, no. Climbing shoes are supposed to be pretty tight.
I wear climbing shoes slightly bigger than my street shoe size, and I still cannot wiggle my toes.
I can slightly move them up and down a little, but not so much that I would consider it wiggling them.
Because we are in pain! But not that much pain. Generally, climbing shoes are not comfortable, and leaving them on for a long period of time just starts to hurt after a while. Taking them off in between some of the climbs just lets our feet relax a little bit, and delays the session-ending feet pain.