Can I run Omnisphere side by side with Atmosphere?

Omnisphere and Atmosphere can be run side by side, since they are separate instruments. Keep in mind that Omnisphere is a completely new synthesizer that also includes the Atmosphere library.

Do I need to uninstall Atmosphere before installing Omnisphere?

No, this isn't necessary unless you are no longer going to use Atmosphere.

All of Atmosphere is included in Omnisphere, so you can uninstall if you like. However, it's useful to keep Atmosphere installed to recall older projects.

Will it replace Atmosphere on those sessions with all of their settings automatically?

Omnisphere will not automatically replace Atmosphere in older sessions. You can reselect the same Atmosphere patches in Omnisphere, since it includes the original Atmosphere library and patches.

Will I be able to open custom patches that I made in Atmosphere?

No. Since Omnisphere is a completely new instrument, it's not possible to open custom saved Atmosphere patches in Omnisphere.
However, you will be able to recreate your custom patches with much more choice and flexibility, and also trade and share patches with other users! This is possible because Omnisphere is based on our own technology, and will allow for song compatibility with future updates, just like all the updates with RMX have been song compatible since it's introduction.