Thievery
##Thievery PartPart of your answer is in the quoted text...
as long as it starts and remains within the 30cm radius.
Unlike other games' invisibility, this effect doesn't apply to things the character picks up after the spell goes into effect - they explicitly have to be on him (or within 30cm) when he casts the spell. Anything the character attempts to steal will remain visible.
A related, and somewhat disgusting, stumbling block is what happens when he eats. If he tries to spend his whole life invisible, he'll have to carry food with him when he triggers his invisibility. What visible, eaten food looks like inside an invisible person is an image I'll leave to others imaginations.
Murder
##Murder AnotherAnother factor is the unreliability of pre-modern timepieces. The idea of him killing somebody and then immediately becoming invisible again requires precise timing. He needs to strike the blow (and kill with the first blow) exactly 24-hours after becoming invisible. Even if he can time it so precisely, there's still a narrow window between becoming visible and being able to trigger the invisibility again.
Of course, even if he somehow did have an exceptionally precise portable timepiece, everything he's carrying becomes invisible when he casts the spell. He may not be able to see a watch, even if he has one.
Summary
##Summary InIn response to your final question, though not an answer to the specific wording... Your player is very much trying to abuse the system. He's not really being all that clever about it, though.