Thank you for your question. You seem like a very organized and energetic person. I hope that I have fully understood your question and please tell me if I miss something that concerns you.
The first thing you need to know is that, with the two of you not being married (I don't suppose you have a registered partnership), the contributions made for your child have no bearing on the house issue. Those are related only to the obligation (which belongs to you both, actually, not just you) to support your son, corresponding to your means. That can actually be enforced against his father, if you want to do that, through the court.
Regarding the house itself, it is true that you own it 50%, but you have a right to claim 50% of your extra costs, with a legal guarantee on the house itself, which means that your can claim those extra costs as a share of the house. Furthermore, this kind of calculation is only academic if there is no dispute. In preparation for a mediation, every kind of argument counts, so yes you should also count the hours you worked on the property. At the very least, they will cancel each other out and you will return to the basics, which is who contributed financially more. Even if the contributions are in fact higher than the house value, what will count is the share you have as a percentage of the house, in case you sell it, or wish to be bought out by your partner.
I cannot see from your question what kind of final settlement you have in mind in the long term, but I can tell you that, as a co-owner (and the person raising the child, making the house a non-family home) you have the right to "exit the co-ownership" which means you can force its sale and receive a corresponding share of the money, or even to buy him out, at a price set by the court, after taking into account your contributions. You can also use the threat of that to force him to buy out your share.
Regarding the bills, what you say is true, for bills such as garbage collection or property taxes. For bills like water, the person using the house should pay them. It is a 'tacit usage agreement'. Use brings responsibilities. On the other hand, it is a good idea to at least contribute 50% as a sign that you are a co-owner, even if not strictly obligated.
Regarding your partner's employment, there is indeed no reason why he should not work or apply for social security. Having never been married, you have absolutely no obligation towards him. As mentioned in the beginning of my reply, he actually has an obligation to contribute to your son, inasmuch as his means allow.
I hope my reply answers your questions and I look forward to your rating, which is essential to my activity.
Dr I L Vlad