Homegroups are a Windows 7 and 8 addition and only these Windows PCs use them. Standard Windows shares are what you need and a PC will not be seen if it does not have any active shares.
Your network problem – I would look to your own network for this – I would think it is because the T2 has received a request to authenticate. I normally setup my shares using the settings to allow read access to my network (your firewall will prevent external access) – so no password required anyway.
You might check the DHCP lease time on your router and the pool size – I would suggest an address reservation on the router for the T2 as well.