I would be hopeful that 12.04 could run on Linux 2.6. It's not much older than the initial 3.0 release so I doubt there's much that relies on 3.0-only APIs.
The core Linux developers still maintain one 2.6 kernel (currently at 220.127.116.11) although in the .58 release notes it was said that the maintenance will slow down and will, one day, eventually stop.
2.6.32.* is also still being built for Lucid (10.04 LTS). You can find the packages for this in the Kernel PPA but if you're reading this answer any significant time after it was posted, check the listings. You will need to manually download the packages for your architecture and install them.
You will also need to monitor 2.6 releases and upgrade under your own volition. Ubuntu won't automatically update these packages for you so you'll have to go back to the PPA.
Lucid support will in April 2015 so I expect these builds to end. After that your remaining options are:
- Download from kernel.org and compile it yourself, though I expect maintenance to tail off after Lucid dies.
- Pay your hardware vendor to upgrade their driver for 3.* kernels.
- Replace the hardware with a version that is still supported.
- If the vendor doesn't exist any more and there isn't replacement hardware out there, you could probably pay somebody to port the driver.
And you have to keep in mind that this might all come to nothing. There simply might be things in 12.04 that rely on 3.* features.
In terms of your auxiliary questions, yes it should be possible to install alongside older kernels and yes, that will result in being offered a choice at the Grub screen. That said the newest kernel will, by default, always be the first choice.
You could probably hardwire around that by editing the grub configuration.