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You would either tunnel all necessary ports (via SSH, for example, tutorials are available via google) or more easily with a VPN. VPN into home and you'll be good to go.

That's not weird, that's how most DHCP servers work. They keep a lease database around that lets them refer to older (expired) leases, so they can assign the same IP address that physical address asked for last time. It's a feature. It's a rather nice feature, really, because while my ISP uses DHCP, since they never drop my IP address is quasi-static. That is, it's been the same for the last 16 months.

Bluesockets are simply Linux-based captive gateways. Unauthenticated you are in a particular role, which allows minimal outbound traffic (but at least outbound DNS). You open a browser, your browser sends a DNS request for your home page, gets the correct result, and then issues a HTTP GET for that page. The Bluesocket controller intercepts the requests and answer it as if it was the other endpoint, and returns a HTTP 302 moved to its own login page. Once authenticated, you are assigned a role. No further connections are intercepted, the role merely governs traffic shaping and firewalling (and an optional redirect to a URL of the admin's choice after successful login). As far as firewalling goes, roles simply contain lines of rules that make up an iptable chain, this chain is applied to your traffic. The administrator of the controller you are behind apparently has chosen to block a bunch of ports and destinations, most likely followed by a catch-all allow any/any rule at the end of the set. This has absolutely nothing to do with your hardware or OS, Bluesocket controllers work entirely based on TCP/IP and and device that can do TCP/IP can be managed by them. Your only option is to proxy all the connections it blocks. Same as with any other firewall.