If I upgrade to iOS 5, will I lose the opportunity to restore from one
of these old backups?
No. You can restore your settings, data, photos, videos, apps, and anything else iTunes has backed up from iOS 4 right over to the new iOS 5. Backups are compatible with future versions of iOS. You could likely restore an iOS 3 backup to a device running iOS 5. The only complication you would run across would be the apps and whether they retained support for your current iOS. Most developers periodically drop support for older devices as time goes on. For example, most new games in the App Store very rarely support the iPhone 3G or older iPod Touches. Moreover, they have also begun dropping support for older iOS versions, such as v3.0. If this conflict every surfaced, iTunes would simply not migrate over any apps that did not support your device (of course still moving over the rest of your data).
Would I be able to restore the phone to the old operating system and
then restore from an old backup?
Yes, and no. Let me explain the "yes." Apple digitally signs every release of iOS. This signature is verified by Apple prior to its installation. This is often referred to as a "signing window." During this window, any firmware can be installed on your phone, provided it has a valid signature and is being permitted by Apple (passes verification). Apple has historically left the window open for about a week and only for the previous version of iOS (e.g., if iOS 5.1 was released tomorrow, the window would cover iOS 5, if iOS 5.2 was released after that, the window would cover 5.1, and so on). More specifically, now that Apple has released iOS 5.0, iOS 4.3.5 (the previous version of iOS) is likely still being accepted, but that window is fast closing (and could likely have already been closed). Additionally, the rollback only applies to the direct previous build (in this case only iOS 4.3.5 and not 4.3.3 or 4.2.1), but it is time sensitive (iOS ups the ante by introducing tighter measures that may prevent the use of SHSH blobs, effectively sealing a downgrade to Apple's small window, but that's another topic altogether).
If you decide the day of release that the new firmware isn't for you, an immediate downgrade is 100% possible. As the days pass by, that number drops dramatically. If you miss that window, you will be unable to install the previous release without SHSH blobs, and software designed to jailbreak devices, which can allow a downgrade indefinitely. It's of note that SHSH blobs must be taken from a device running that particular build. You cannot extract SHSH blobs for iOS 4.3.3 from iOS 4.3.5.
So it's a mixed bag. Yes you can downgrade, but you must act fast. After that, it is a resounding "no." Unless of course you have met the requirements (SHSH blobs, etc.). However, as I said, iOS ups the stakes by seeking to block (or rather change the behaviour of) SHSH blobs, effectively limiting a downgrade to Apple's window.
This of course omits discussion of the baseband (the component that handles cellular data) which is another factor to consider should you requiring unlocking the handset to handle multiple carriers, however, this falls slightly outside the scope of your question, and for your intents and purposes, something you needn't worry about.
Best Answer
You may be stuck on iOS 7 if you don't have a way to get that software from Apple's servers. Typically only registered developers can download several versions of software, but if you search - there are many places that archive these software ipsw files. It's always a bit of a risk downloading software from a place other than the vendor, but I don't know of a better option.
Typically, when a new OS is released, iTunes will only download the latest software if you don't already have an older version of the software stored locally. Is there any chance that computer (or another computer) downloaded iOS 6 and still has it on the hard drive? If so, you can decline the offer to upgrade and restore the device from that version of firmware.
As was covered in the comments - there is no restriction to load iOS 6 backup on iOS 7. Also - this is more about the computer and iTunes having the old iOS file already downloaded than something that's inherent on the device. Any change you have a backup copy of the iOS files iTunes downloaded?