I have heard about studies that show second/third/etc children's motor skills develop quicker in an effort to keep up with their older sibling(s). Is this also true for eldest children in a daycare setting with older children? Are there studies to support this as well?
Children who are routinely challenged develop more quickly in the areas where they are challenged.
A day care may or may not challenge your child. They vary incredibly in both programming and quality.
If you are concerned about your child's motor skills, get them used! Throw a ball, play hopscotch, play an instrument, study martial arts, dance, whatever keeps him or her moving.
Kids know the difference between adults and others kids. As much as they will try to emulate adults, they are more fascinated and interested in other kids. Be it day care, Sunday school or neighborhood kids, kids will learn from older or more advanced kids, especially when they become at an age when they are actually interactive and want to play with other kids.
For my son's case, when he was in a multi-age home daycare (from 13-18 months), he learned to feed himself (combination of not having 1:1 adult attention and observing other kids) and "cook" as they had a large play kitchen (everything from handling pots & pans, opening up the oven and various shelves). He wasn't really every playing with the others kids at this age, he'd just copy them and do his own thing.
If your day care separates kids out by age, you may find less of this modeling, since the kids will all be of the same developmental stage.
Rather than motor skills day care helps in improving independence, taking care of oneself and sharing, adjusting qualities. Since they have to share toys, they have to adjust to the other kids and they have to spend a large time of day without mom / grandparents who constantly fulfil their wishes, the qualities mentioned above improve. Also may improve their verbal skills since they get lot of children around to practice talking.