How do young Catholics remember a set of prayers in order to pray the rosary? Do they have to memorize everything by rote, or do they get some help with mnemonics?
Rosary is a combination of many prayers like Apostles' Creed, Our Father, Hail Mary, Glory to the Father.., twenty mysteries divided into four sets, Hail holy Queen, Litany of Mary etc. All these individual prayers are therefore memorised so that one can say them fluently at the time of their recital. All these prayers are mandatory (to be learnt by heart) before the members are made eligible for Sacraments like First Communion, Confirmation etc.
Should they pray in Latin or in the vernacular?
All these prayers are said in vernacular languages. In my place the Litany sometimes is still recited in Latin, but that is by a few exceptional groups of people who are of older generation.
Is there a bit of wiggle room if someone is a little amnesic so that they don't have to remember a set of prayers?
I am not very sure but for such members only the basic prayers like Apostles creed, Our Father, Hail Mary are made to remember so that they become eligible for requisite Sacraments.
Would just reading the prayers suffice instead of actually memorizing, or is memorizing a form of pious behavior for young people who are taught to become faithful practicing Catholics?
It is necessary and mandatory that all these prayers are memorised and this is usually done at home as well as in the Sunday catechism classes. There are about seven text books that children have to learn and pass during the catechism classes. During these classes they are taught these prayers as well as other tenets of Roman Catholicism/Christianity. But mostly the memorising is done at home. Then they have to answer exams of sort in the catechism classes and pass them. The Sacrament of First Communion is given for those who have learnt and passed first two text books. And those who have completed all the seven would be selected for the Sacrament of Confirmation.