The healthcare coverage costs seem like a low blow, and may be negotiable since it wasn't mentioned earlier in the process. However, if every other employee gets the same deal out of them you'll have to decide whether you wish to opt out (you've been without one so far, so maybe you don't really need it), or set up a separate package yourself (if these are valid options).
If they aren't and this setup is compulsory you may argue for the amount to be paid by the company, but if all their employees get the same deal then you may be out of luck.
As far as this goes, you never know until you ask. Be polite, and honest:
Once the costs of healthcare and day-care are totaled up I actually end up making less than I am at my current job. Would you consider offering me a one-time signing bonus?
You shouldn't make it conditional, or imply that you won't take the offer unless they do, however don't flat out say so either. Let their imaginations run wild. The worst they can do is refuse. However you may be surprised.
This may seem a little harsh, but I'm just going to be brutally honest:
As far as the cost of day care is concerned I don't think you have a leg to stand on.
An employer offers a salary based on skill, experience, education, and value to the company. You having to take care of your 3 yr old is not their responsibility, it's yours. How you manage to do that - whether you enroll the toddler in day care, ask a relative to baby sit, etc. - is really not their problem, and I personally don't think it constitutes a valid bargaining chip unless you're incredibly valuable to the organization (and even then it's more of a strong-arm tactic than anything else).
If they were to give you a higher salary based on the fact that you have high day-care costs then those employees who don't have a child and make less than you would feel slighted, and maybe even discriminated against. Furthermore, other employees may come forward claiming various costs.
An equivalent question (to my mind) is whether it's reasonable for the company to pay for my commute simply because it's expensive. The answer is no, not really. I'm expected to be at work on time, and how I do so is my business.
That may sound harsh, but it's most likely how their management team is going to view the issue.
Asking them to allow you to work from home is risky since you've mentioned that they are generally against it at the moment. As I've mentioned, you may or may not have an ace up your sleeve depending on how much these people need you, however I would hesitate asking for too many privileges because at one point you're going to be far more trouble than you're worth.
I think you really have to decide which of these jobs offers the better long term prospects for you and make a decision based on that, rather than ask for too much up front and risk losing the offer.
If the new job offers you a better opportunity to gain experience, upwards mobility, or raises further down the line then it might be a better fit than a job which allows you to work from home, but which doesn't challenge you and which doesn't allow you to grow professionally.
I don't know that you're ever going to find the perfect job (great salary, day-care allowance, work from home perks, etc.) - few of us ever do - however I wish you good luck!
Note: in the future I would not reveal what you currently make, but rather what you're interested in making if you were to accept the job (plus $5000, because they will always low-ball you anyway). That's what I do in my negotiations, and it's worked well enough for me (although different people take different approaches).