From what I understand no backup is safe online at the moment... which was always the case really. Microsoft has already shown they can and will flash the firmware of the drive with updates. So, I'm sure they can deal with the modded firmware whenever they want.
As far as your question about playing on LAN (non AP25), if you aren't connected to Live, then I see no reason why you can't. As long as the backup is good and verified you should be fine. From everything I've read and experienced there has to be some kind of flag thrown due to the disc you are using, which would trigger a ban when you come online next time. I personally played MW3 truncated version 2 days into the silent dash update and luckily never got updated or banned. As soon as I found out I pulled the ethernet and just played offline. I have cleared the cache since and have done a XBLive connection test and all was successful with no bans. I also tested using Xval and I have no flags showing pending ban either.
Previously I had a non verified burned Gears of War 2 that I had played with no problems. One day I put it in my second modded xbox, which had NEVER been online with it and played a bit. The next time I plugged in the ethernet and connected to live I was greeted with a ban! So, long story short, as long as the game is verified and correct, you should be fine assuming you are using the most recent firmware online. And if you are playing offline make sure you have a verified disc and you should be fine.
Lastly you asked if you can burn with any DVD burner now. Yes, and no. You can truncate MOST discs (not all from what I read). But, these are obviously NOT perfect 1 of 1 copies so you run the risk of getting banned. To burn all data you need a new iHAS Liteon burner. There are different kinds (A and B versions), so make sure you get the right one! They are under $20 shipped so worth it. (Mine came 2 weeks ago and is still in the box!)
I finally just bought an old ELITE system that was unmodded for $50 that I am using for online being as I only play MW3 or a game or two online each year. I will play everything else offline on my modded box. Seems to be the way most people are going nowadays. Much better not having to worry about all the updates and whatnot.