No. The second you do that you will be murdered with all your belongings in the first nullsec system you dive into.
Also no. You go to the recruitment section of the forums and take a look. Most corporation which are worth joining will advertise that they are in a nullsec alliance.
Before you join such a corporation you need to make sure that the alliance this corporations belongs to is close enough to what you want to do and or gives you room to grow as a person or in general.
Most alliances will require you to do things you really don’t wanna do and also require you to know everything about that task.
Some but not all alliances will teach you things and or give classes for things you can do or are interested in.
When you decide on an alliance to join, make sure you also consider how you will be making isk, since no matter how many sooper doopers they may or may not have, you will lose ships often and since flying space ships is usually the coolest thing ever, you will need to replace your ship at some point.
The way you like to earn your isk should determine if you join a more industry or less industry like corporation but most corporation are very lax on that kind of thing.
However if your soon to be corpmates are more centered in doing mining ops for making isk and you would rather shoot down red thingies for example, your nullsec experience may not be a enjoyable as it could be.
If your soon to be corpmates are more into piracy and drop every freighter in high or lowsec they can find and you would rather earn your isk in a more honest way, your experiences may not be as enjoyable.
Nobody can tell you ‘what to like’. Only you can decide what you like.
Once you figure that out, head to the recruitment section of the forums and ask for a corporation.
Look at the existing threads and tell people what you are looking for.
If you just wanna day-trip into nullsec without making any commitments, you should keep an eye out for ‘spectre fleets’, which you can find in the fleet finder.
Those usually start in highsec but will venture into nullsec and are open for everyone, no matter which corporation or alliance they are in.
I suggest you try a spectre fleet as a test-run into ‘fleet-action’, so you can get an idea what to expect, without having to make EVE your job you pay for.