Is NTFS on Ubuntu stable?

It's because almost every distribution uses the NTFS-3G driver:

The NTFS-3G driver is used by over 240 distributions. Many, like Fedora, Mandriva, openSUSE and Ubuntu with over 8 million users, use NTFS-3G as the default NTFS driver.

=> NTFS driver for Linux

And yes, it's stable :-)

Do you know why I can't see an option to format partitions like that while installing?

Do you mean the access to your Windows partition from an already installed Linux OS, or do you want a fresh Linux installation? There should be an option to erase/format your old (NTFS-)Partition/s while installing a fresh Ubuntu/Fedora/...

Rule of thumb: do not rely on NTFS if you use Linux. Lots of claims that ntfs-3g is stable. This is totally wrong.

NTFS is a closed filesystem. There are no open specifications of it. Everything that ntfs-3g can do was achieved by reverse engineering. So if it works reasonably well in many scenarios, this is not a reason to consider it stable and reliable.

No, it works out of the box in Ubuntu 9.04 and it is considered stable. It won't let you mount partition if Windows is hibernated though but you can do that yourself from command line only please do that in read only mode.

Yes, the NTFS support in Ubuntu is stable. ntfs-3g is the standard driver used for NTFS access in linux.

However, you should be aware of one point before you use NTFS on linux:

Note that all three userspace drivers, namely NTFSMount, NTFS-3G and Captive NTFS, are built on the Filesystem in Userspace (FUSE), a Linux kernel module tasked with bridging userspace and kernel code to save and retrieve data. Almost all drivers listed above (except Paragon NTFS for Linux) are open source (GPL). Due to the complexity of internal NTFS structures, both the built-in 2.6.14 kernel driver and the FUSE drivers disallow changes to the volume that are considered unsafe, to avoid corruption.

The above is taken from the Wikipedia NTFS page. I ran into this issue once, where some operation accessing the NTFS drive hung, and I had to terminate the call. From then on, I couldn't get the NTFS drive mounted on Ubuntu as it kept telling me the drive was not in a fit state to be used. I had to attach the drive to a Windows machine and boot up into Windows, and then the drive could be mounted on Ubuntu again.