The second network has no management network ...
The Management Network is used to manage the ESXi host itself and has nothing to do with the Virtual Machine networks, so there's no need to setup a Management Network on each vSwitch.
... and I have no internet on the windows 7 vm.
How could you, without an uplink. It's basically the same as with a physical network. If a PC is connected to a switch and there's no connection to the Internet (e.g. to a router) you won't have the chance to work online.
Will it show up and could I bind that to VM Network 2 when connected so that the server can service physical clients as well?
Yes, this should work.
Do I need to bridge the connections somehow to allow the win 7 client to access VM Network 1's internet connection.
What's the purpose of separating the domains if you now want to bridge the networks? What you may do is to use a firewall (e.g. pfSense), configure a separate vSwitch for each domain (without uplinks) and setup the firewall to connect to each "domain" vSwitch as well as the one with the uplink to allow Internet traffic for the different domains but block direct traffic between them.
André