For a small fee (well, ~600 Gold), one Prelate Sabinus found at the back of the Mages Guild in the city of Evermore will cure you of your condition.
It appears I have gone and created more questions then I have answered.
First of there are 3 ways to level up once you hit Veteran Ranks; Veteran Overwold zones (questing), Veteran Dungeons (same dungeons you did while leveling but much harder) and PVP.
The short answer is more like your 3rd option. Not in the way you described it though. Level 50 A.K.A. Veteran Rank 1 is when you are sent to the Veteran Overworld Zone, which is the Veteran questing area where you do the questlines of the other two factions. Only in these areas everything is Veteran Rank. Veteran NPCs, Veteran quests, Veteran Rewards, Veteran Gear, Veteran level resources, you name it, its Veteran.
If you so choose to go back to your home faction questing area the highest loot you will find is level 50. Nothing will be Veteran except for the occasional resource you might find in coldharbour that lets you make Veteran gear. And yes you can go back simply by teleporting.
Also the first time you do this, (speculation at this point) the questing will only put you between veteran 4-6 which you will then start the next factions questing zone. The idea behind Veteran ranks is to take a very long time.
Alright time for my long answer full of spoilers and such (this is my attempt at not causing more confusion)
So you have finished Coldharbour, foiled Molag'Bal's plot of destroying Nirn. You have just hit level 50 (AKA Veteran rank 1(hopefully, those that don't have to grind to 50 cringe)) Now you are being summoned to the harborage to fight molag'bal. Once defeating him you are asked if you wish to see what it would be like if you had washed up on a different shore, saved by a different faction. If you accept, you will start your journey through whatever faction the game chooses for you. I am unsure if it is random or if there is a linear path of which one is next. You start out just like if you had started out making a new character for a seperate faction, except you are still veteran rank 1 and still have all your skills and gear. Also everything is veteran as well as mobs which by the way are much harder to kill. Traps are just as deadly, one misstep will hurt, two missteps will be deadly. So if you couldn't help yourself and read this part of the answer enjoy veteran content its way more challenging and fun then normal mode.
Best Answer
This can happen for several reasons:
I have found (experience and logic) that these cause quests to not be 'shareable'. The share button will still be enabled but no notification of any kind is sent to the group or yourself. I have no reference for this however.