Personally i think the problems here lie with the business, the other problems you mention are general day-to-day problems faced by every relationship.
Responsibility should be shared in a relationship, especially when children are involved. Do not make the mistake of thinking that this should be the case with your business or you will fail and it will ruin your relationship also.
If you are going to do business with your partner, then you need to be very clear on who is responsible for what, and remember its very difficult to 'discipline' someone in the workplace for not fulfilling responsibilities when that person is also your partner.
Run your business like a business, not a family.
1. If you take cash you need a safe.
Any cash that is collected should go directly to the safe. Any other cash collected should go into a business account.
2. Have a paper bin.
Your office should never be so untidy that important documents could get mixed up with anything else.
3. Give yourself both very clear roles.
Don't just share the workload, no real business works like that. You need to set up departments and roles, and fill them between you.
Can i just ask too, did you tell him there was £2000 in an envelope before he cleaned the room? If the money is gone, it's gone. Placing too much blame will have no use (especially if you had a part to play) other than to break up your business and possibly your family.
Based on what you say though, it sounds like you could have quite a profitable business which is better than a lot can say - look at the good points where possible, never make the same mistake twice, and be organised. Either that or give up on the business and work out if you want to make the relationship work instead. I think ending both the relationship and business at the same time would be pointless as your judgement will be clouded.