Objective 'win a gift in a contest'

I don't know for sure, but It may mean getting a prize in a promo - have you tried making a prize in the current merge event? (Transformation: Winter Menagerie)

I did try go into it but it just tells me that it isn't available right now

You may have to wait for the next promo - I think access to promos depends on seniority, but if so, I don't remember how many days you need...

If I remember correctly I could start right into the promo when I created this account. Maybe it's something about playing mobile? Or some tutorial from the promo not being finished?

The creation space is restricted until you reach 60 (?) days of seniority.