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.
Long story short, yes you can make 50 on just PvE.
I am currently half way through Veteran Rank 1 and I had no problem getting to level 50 with just quests. It was close I will say, being on the final few quests before finishing coldharbour. I will also mention that a lot of people are having issues with finishing coldharbour and only being level 45-49.
I pvpd maybe once during my journey to 50. I did the beginning questline that gives away 2 skillpoints, and then continued to finish the 20 kills quest. so maybe a small 30-40k boost of experience, so nothing huge. I did every quest I could find, only not finishing Shadowfen (by 7 quests) and The Rift (by one quest) because I couldn't find them. Also missed one quest in Coldharbour because it is bugged (I keep going back to it every morning as this seems to fix most bugged quests, I also wake up at 7 am everyday so it is pretty quiet and I think the servers don't freak out as much.) I know I missed these quests because the achievements tell me so, although I have noticed in Deshaan and Stonefall had more quests then the achievement said there were.
I would say a lot of peoples issue with hitting level 50 on times is that they only do 70% of the content. As Exploring, public dungeons, private dungeons, and side quests is atleast 50% of what you need to level on time. While the main storyline quests and the quests you happen to walk by doing the main storyline is the other 50%. So if you are worried about not making 50 on time make sure you are doing atleast 90 - 95% of everything so that you don't find yourself spamming coldharbour for a grind party.
I will finish by saying I also found myself multiple times 1-2 levels below the mobs and quests, it was confusing me a little at first but do not worry. Take your time, enjoy Tamriel and remember to take some time to just explore, discover and finish some things outside the beaten path.
Best Answer
Yes, the game will allow you to finish both, although at a slower pace.
The passive ability only dictates how many research jobs you can actually start while others are still in progress and how long they take.
So if you respec the timers will increase, but that's essentially the only downside of not getting those passives again (aside from not being able to start multiple jobs).