Yes, you do need Internet at home. It'll come handy, especially since you're at a key time in the progress of the company. However, 5% doesn't seem exactly as a fair deal to me. I know it's OT, but maybe you should consider negotiating your terms and equity in the company, even more so since you're the one taking care of the systems and scalability.

Also, everyone needs a little decomposing time, so cheers on that initial part of your post.