No, there is no capitalize when referring to rosary beads. You may choose to capitalize when referring to set of prayers called the Rosary, but not for rosary beads.
By the prayers of the Holy Rosary.
I believe they are called a rosary.
Rosary
The Rosary was developed as an aid to support the faithful in their prayers. Specific prayers are not so much assigned to each bead as there is a bead to mark each prayer.
The Roman Catholic Church and the Orthodox Church are the two main religious institutions that pray the rosary, though other Christian denominations and even other religions pray the rosary.
Roman Catholic AnswerThe link below has the prayers of the Rosary in Latin and in English.
Dominic de Guzman did not 'invent' the rosary but did promote it.
The Rosary is a string of prayers so since you say "at a catholic rosary" I assume you mean this is a public event where the Rosary is said. There will probably be one person (probably a priest) leading the prayers and everyone else joining in for the last few lines of each prayer, like a chant. It could also be in latin.
It's the other way around, the physical rosary was put together the way it was so that you would be able to see where you are in your praying of the Rosary.
The rosary helps Catholic Christians to concentrate on sacred mysteries and provides them with a formula of prayers when they may not otherwise be able to find their own words to express their prayers. The Catholic Church has also heavily indulgenced recitation of the rosary, especially in a family setting, thereby granting special graces to those who pray the rosary.
J.M Cruise has written: 'The rosary without distractions' -- subject(s): Catholic Church, Prayers and devotions, Rosary