I pulled my deltoid muscle. what do you recommend for pain, hot or cold? and, would massaging damage more?

Either: Ice for first 48-72 hous. Heat after or alternating heat or cold after.

Ice first, then heat: You could ice it for the first 24-48h to keep any swelling down, then heat and warm compresses can be used to further provide relaxation to the affected muscles and comfort. Nsaids could help. Massaging may help, but let pain be your guide, if painful, then tone it down. Seek help if not eventually improving. Get well.