Yep, tie them in a pillowcase and go for a cold, gentle cycle.
I use this stuff from Lakeland made for laundering delicates, but Woolite or even a squidge of Johnson's baby shampoo will do.
Perfectly okay to dry on the radiator, but if the weather is good I have pegged cuddlies out on the line by the ears/tail, etc.
Something to keep in mind is that babies and kids can vomit quite a bit, so more special cuddlies should be for playtime as they get older rather than in the bed/cot with them (for older babies and children, of course).
I've laundered things with music boxes and such (see 'babies and kids can vomit quite a bit) and as long as you remove any batteries or bits in general it's fine.