1a Modded : Play Original = Will be banned if caught
1b Modded : Play Original + Online  = Will be banned eventually.

2a Modded & Browse content = Banned too, eventually
2b Modded & Download demo or video = Banned too, eventually

3a Depends if your xbox downloads the update released on 20th October 2009, which is probably going to be downloaded into your console the first time u plug it in to Live.
3b Yes if you have downloaded the update. Some people has escaped Gamertag ID ban by having 2 gamertags on console. The ban results in gamertag id ban, save games crippled, install hdd removed, watching video disabled. Many more unknown. Carrying these items to an unbanned console would restore some features Gamertag recovery is neccesary then

4a Connecting to Live already tells MS your serial, which also allows it to know its Jasper.
4b. Yes they know immediately.

Conclusion :

You will be banned regardless of what you do. Trust a person who has read 250 pages of modding the xbox
It doesnt matter whether you play your games earlier than release date
It doesnt matter whether you play your games after release date
It doesnt matter whether you play original or pirated
It doesnt matter whether you installed to hdd or not

Fact is real that MS knows you modded and what ever original you try 2 do after that just flags you, making you wait for a ban to be sent down to your console.

When you are modded, just connecting to MS servers would treat u as unmodded console... until they decide to check it throughly.. MS are very confident ppl when they ban and when they have found enough evidence. They dont revert their ban hammers for anything. Small groups of ppl however has managed to find a loop hole to their ban hammer.. I wouldnt share about that.

Yes, it is unavoidable. As long as you have connected to Live with it, MS knows you are out there and would probably just flag it for banning, this year or the next.. Some sooner, some later..