Simply power it on. The new OS will automatically update each file as it boots.
Android 4.3 OS (Jelly Bean) should already be available for the Galaxy Nexus.
Under Settings > About Phone > System Updates, you can update your Samsung Galaxy Nexus to the latest Android OS version, currently codename Jelly Bean (4.2.2 or 4.3 depending on the region and Nexus variant). I would suggest updating this over a wifi location, as the files tend to be large.
Nothing really "happens". You still have the same phone but just not Android Jelly Bean. Your phone company may release an update for you or a custom ROM may have been created online.
If you are asking what OS the Nexus 10 uses, it is the Android OS (4.3--Jelly Bean).
Yes, the Sony Experia should be able to upgrade from Ice Cream Sandwich to the Jelly Bean upgrade. They started the upgrades in mid 2013.
Both phones are manufactured by LG, and they largely have the same architecture. Some of the differences is the G runs on an LTE network (Nexus 4 is HSPA+), has an expandable microSD slot (Nexus 4 has no expandable slot), has a removable battery (Nexus 4 battery is sealed), a 13 megapixel camera available (Nexus 4 only has a 8 megapixel camera), and runs on Android 4.1 Jelly Bean with a manufacturer skin (Nexus 4 has Android 4.3 Jelly Bean stock).
A company that makes jelly bean machines and dispensers is the popular jelly bean producer Jelly Belly. Another company that makes a jelly bean machine is The Jelly Bean Factory.
Check under software settings to make sure an update isn't already available. If there isn't search around for a developer created ROM running Android Jelly Bean that you can download. Keep in mind that these are not official and might brick your phone. Read all the instructions before you download.
You won't have to root it to get the update, but if you want to keep your root, then you'll more than likely have to redo it.
It depends on your needs. Both phones have very similar chipsets and processors, same memory, same basic storage (neither are expandable), same rear camera. The DNA has a larger display with higher resolution (5 in/1920x1080 pixels vs. 4.7 in/1280x768 pixels), has LTE (Nexus 4 has HSPA+), better front camera (2.1 vs. 1.3 on the Nexus), removable battery (Nexus 4 is non-removable), and older Android OS (4.1 Jelly Bean vs. 4.3 Jelly Bean on the Nexus).
The Jelly Bean Wallpaper was given out on 'National Jelly Bean Day'. Watch the Daily Growl for 'National Jelly Bean Day'. Moshi Monsters may or may not make the jelly bean wallpaper available again.
Jelly bean