[RPG] Can a monk with Empty Body break the invisibility while keeping the resistances

Yes

The spell Disguise Self is an illusion. It does not actually change your (invisible) form and the description points out:

If you use this spell to add a hat to your outfits, objects pass through the hat and anyone who touches it would feel nothing or would feel your head and hair.

and

If you made yourself appear thinner then the hand of someone touching you would bump into you while it was still seemingly in midair.

From the description this illusion is obviously visible - cloaking your actual form - and, like any spell, your own state of (in)visibility doesn't matter.

This is not that different from casting Silent Image that "creates an image of an object or creature or some other visible phenomenon". This would also be visible regardless of the caster's visibility. Disguise Self is doing pretty much the same thing, except the illusion is restricted to a roughly humanoid form that is superimposed over your own body.

So I would say 'yes' you can render yourself temporarily visible with this spell.

Thievery

Part 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

Another 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

In 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.