The usage of 醒来 vs. 睡醒, and a little bit about 睡觉。
醒: wake

醒来: wake up

睡醒: wake from sleep

睡觉: (go) sleep

I am a native Chinese speaker, IMO 我想在你的身边睡醒 and 我想在你的身边醒来 or 我想在你身边醒来 (no的) sound equally fine.  我想在你的身边醒来 is slightly more spoken expression, while 我想在你的身边睡醒 is slightly more written.  

睡醒 and 睡觉 do have opposite meanings, but a better antonym for 睡醒 is 睡着，which means "fall asleep".

睡了 and 睡觉了 both sound Ok to me to express "go to sleep".

in 睡觉, 睡 is actually the verb "sleep" while 觉 is the noun "sleep". I think 睡觉 become mainstream just because nowadays the chinese spoken language like to use phrases with two phonemes.