Is it normal for me as an asexual to want to be in a relationship?

Sexual? No.
Romantic? Sure.
Friendship? Yes.

This is normal, not all asexual people are aromantic. Not all romances involve sex.

I heard a quote once, it’s about being a virgin but also applies to being a celibate asexual, “You shouldn’t feel bad that you’re not having sex, most old people aren’t having sex either so really you’re just ahead of the pack”.

That depends, are you truly Asexual? As in never really wanting / needing sexual satisfaction or is it just for this relationship?

A real and good relaionship is first a friendship. I have a variety of them. Only a few ever became sexual, which usually is a different kind of relationship.

Your feelings are totally normal.

Go for it!

Yes. You can be in a romantic relationship without having sex. If you and another person are in love with each other and you do all the relationship-things besides having sex then that's a relationship. That's up to you to define, not other people.

Depends.

Es tu sûr qu'être en couple c'est la même chose que être ami ?
Demandons nous et apportons nous le même soutien à un ami qu'à un amoureux ?

we all need a relationship of sorts, but you don't say what type you want?

It's more likely you were born in a messed up age and started identifying as asexual than you actually are asexual.