Do you think I should sleep with my splint on my arm?

Probably. It does depend on why you have a splint there. If you are wearing a splint for a chronic problem (like carpal tunnel), i'd wear it at night, so you can heal faster. If you can't sleep well with it, try a more comfortable one. If it's for a temporary problem (like an injury), you may be able to get by without it after the first 2 weeks as long as it continues to improve.