If your TV & broadband are both failing at the same time, there may be a common fault with your incoming cabling, but that's also far from proven and may not be the case.
Some of your error codes may also be caused by the connection between your 360 & homehub - so there's quite a few things that need sorting here. First thing - was your install a self-serve QuickStart, or was it a manned engineer one? The signal levels should have been checked if it were the latter.
How is your 360 connected to your homehub, that's how it connects to the internet for OnDemand & streaming. This can either be Ethernet cabled or wireless - how's yours?
And although this thread will focus on your TV issues, when you say you "keep losing broadband" - does this affect Ethernet cabled connections or only wireless ones? There is some further help that can be given with this, but the thread might get very complicated if we haven't at least covered the basics first.
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