Ubuntu – Dual boot UEFI Windows 7 and Ubuntu 12.04 (both 64 bits). W7 entry doesn’t appear in GRUB

1. In theory, no, you don't need to disable Secure Boot; Ubuntu 12.10 includes Secure Boot support. In practice, this feature is still new enough that it may not work reliably on all systems.
2. No, you don't need a second ESP; Ubuntu should be able to share a single ESP with Windows (or any other OS). In fact, sharing an ESP is the way it's supposed to work (although multiple ESPs are also legal).
3. The "boot flag" is simply libparted's (and therefore GParted's and parted's) way of identifying an ESP.
4. In principle, you can mount your ESP anywhere you like, or not at all. In practice, some tools may require the ESP to be mounted at /boot/efi. This would be most important for GRUB package updates, but some other tools might need this, too.
5. In an EFI system, GRUB doesn't install any code directly to the hard disk's MBR, although of course code does go on the hard disk -- just in the ESP. Thus, installing GRUB on /dev/sda doesn't really make sense. That said, some user interfaces may refer to doing so because they haven't been properly updated for the EFI case. (I'm not sure what Ubuntu 12.10's state is on this score, offhand.)

You've booted in BIOS/CSM/legacy mode, so the installer is trying to set up a BIOS boot loader, not an EFI/UEFI boot loader.

I recommend you go back and find a way to boot in EFI/UEFI mode. One thing that might help with this is using an EFI-only boot medium. Unfortunately, the Ubuntu installation discs are Frankenstein-monster conglomerations of half a dozen more-or-less incompatible different systems that are made to work, after a fashion, through technical trickery. The trouble is that, although this sort of thing works much of the time, it fails sometimes, and when it does fail it becomes harder to disentangle everything.

The easiest thing to try is likely to be to create a second USB flash drive with my rEFInd boot manager. Insert both it and the Ubuntu boot medium in your computer and tell the computer to boot to rEFInd. If you're lucky, it will present a boot option for the Ubuntu installer and that will work. I can't promise that luck will be with you, though; you may need to go digging through firmware boot options or take some other radical steps to get it to work.

Oh, and you may also want to read my page on doing EFI-mode installs. It doesn't directly address the problem you're having, but it covers a lot of potential EFI installation and boot problems.