Lastly (and back closer to topic), I would point out that to Bitcoin a sidechain should look much like any old user. It is no more 'storing Bitcoin information outside of the Bitcoin blockchain' than it would be keeping one's private keys private. To the extent that it is, that is a good thing in my mind. There is utterly no need to document your purchase of a hamburger in a persistent (forever) way which is replicated on multiple copies of the blockchain and distributed around the world. Cool as it may be, it costs money and limits flexibility. It is also something of a privacy issue to be honest.
I think a gift card is a good solution to this. I also think that using a theoretical (not yet possible based on all I've read in this thread) sidechain would ultimately use just as much data overhead to send the Bitcoin to the same places unless you actually bought hamburgers in two separate instances at the same establishment before said establishment converted back to Bitcoin. The only way around that issue is to give someone or something else control of your Bitcoin private key. We've already discussed the possibility of a trustless system that could resolve that, but even then, it would effectively be a mixer, and potentially regulated accordingly.
ETA: Also, the trustless system presumably has less hashpower, so is less secure. Moreover, cash would be another good way to work around the hamburger issue, and ATMs are even popping up to deal with this, but I'm assuming you want a sidechain that is effectively denominated in BTC either because you don't want to carry cash, which may allow for the gift card option, or you are imaging a future where fiat isn't a thing, in which case a gift card could be denominated in BTC.
1) The idea of sidechains is that many millions of relatively low-importance transactions could, and usually would, be aggregated into one bitcoin transaction. I might send one Bitcoin in one transaction to the sidechain peg of some sidechain dedicated to tipping. I would then perform micro-transactions within the sidechain to my heart's content. On the other side, a person being tipped would probably rarely bother to cash out into Bitcoin (or a different sidechain.)
This is not very different to what we have today with exchanges where huge amounts of trading happen 'off-chain'. A difference is that with a decent implementation I don't have to trust the sidechain and I can verify to a reasonable degree that this is so (just like with Bitcoin proper.) This is not at all the case with non-backed solutions and because of this they regularly steal and probably inflate the currency as well. The user basically sees a display transmitted out of a black box...an hears all kinds of marketing about how honest to operator is of course.
Ultimately all this does to Bitcoin is to make it seem like there are mostly large players doing important and high value transactions (and thus capable of paying generous transaction fees) where in reality a universe of people are reaping the rewards promised of a solution such as Bitcoin. This is no loss at all to Bitcoin and I think it is a giant benefit because trying to service the world's exchange economies cannot help but balloon the system to something which doesn't really resemble the original almost at all. Not only that, but a 'batch mode' system like Bitcoin is inherently disadvantaged for real-time operations as are most burger sales. Sure there are bandaids but they are kind of ugly in my opinion, and ultimately result in provoking the same non-trustless system constructs that we see in the current crop of non-backed off-chain providers.
2) Yes, 1 billion transactions is 1 billion transactions. The point is that there would be many many sidechains which I don't give two shits about. If I am acting as an infrastructure provider (by running a node or whatever) most of them don't effect me. I support the chains that I need and care about, and those which have not grown to a point that is uncomfortable for me. I'll be more than happy to support the Bitcoin core as well if I can. That's why I'm so keen to avoid having it bloat to the point where doing so is no longer realistic.
Edit: And because of the peg I don't have to worry about throwing money down a rat-hole when I choose what sidechains I like. If it fails I can get my money back (again, if properly implemented and I do my due diligence correctly.) That is one of the biggest reason I've not messed around with alts which, in principle, I've never had anything against even if they do draw interest/value away from Bitcoin. Sidechains give the benefits I like in well implemented alts without those two disadvantages.
3) Not sure what you are getting at with 'gift card' but in most mainstream implementations they are highly centralized...as are most mainstream things generally. Some sidechains might (and probably would) work on a sort of a token model. Those could be considered micro-gift cards of sorts, though simply one's footprint in the sidechain would also.