Well... they don't block "all" outgoing ports, since you wouldn't be able to connect to any websites or even check your e-mail if that were true.
If your server doesn't work properly on ANY ports (including the high-ports used for temporary connections, like 10567), then it is possible they are inspecting all incoming headers to check for initial HTTP connection headers. If that is the case - there's nothing you can do to bypass it aside from using using a VPN or tunneling.
Aside from that - hosting anything off your personal computer that will be available publicly is always a bad idea. Web servers like Apache have been time-tested for years and years, but they
still have severe vulnerabilities.
Anyone who runs a public server know all too well how often people attemtp to break in. The
recent case with Apple once again proves that even multi-billion-dollar corporations can get attacked.
It isn't worth putting your personal network, let alone your personal computer, at risk. Test & dev on your local machine - but don't ever open it up to the 'net if there is anything on there you don't want to get out. This is why banks and ecommerce companies store encryption keys and databases on servers that are not directly connected to the outside.
Bots will also often scan open servers and slam them hard if they find common ports open, especially on residential IP blocks.
As SrWebDeveloper said, pay the $3-5/month and get a cheap host. Everyone will be happier.
P.S - If you really want to experience setting up a server from scratch, you can get a
cheap VPS for a couple of bucks per month. They are a great way to learn, without risking your data. As a bonus, a lot of them have sick speeds that will put your home connection to shame