Welcome to Middle-earth!
Absolutely! Unlike e.g. Necromunda, the core rules you play with are exactly the same. You only need a handful of figures to play, and slowly expand your force during a campaign. At the end, you'll have the start of a small normal force. Playing with a dozen figures on the table allows you to learn the basics without being overwhelmed by special rules and abilities. It's also just really enjoyable; one of my favourite options to play the game.
2) Are there any Battle Companies I would be able to build using just the Pelennor box contents?
, you start with about 4-10 figures for most factions. In many cases, those will be the basic plastic warriors. Over time, you will get reinforcements (mostly more basic warriors) and gain promotions. The latter will mean you will eventually need some elite choices (which is useful from a force-building perspective, as you slowly build up a fully-fledged force). You definitely have a great start for Rohan, as you start with infantry and will need some cavalry after a couple of games. Army of the Dead I don't know from the top of my head, but I imagine they just start with a couple of basic infantry. Mordor not so much; even proxying them for normal Orcs barely works as they have different wargear. However, I think you could proxy them quite nicely as Gundabad: heavy armour, some shields and spears, and I think they can hire an Ogre which you could proxy the Mordor Troll as. The minor heroes that lead the band are often just represented with basic soldiers, but you could use the Théoden as the leader for Rohan of course.
3) If I would need other models/units to get started, what would be my next purchases (besides the Battle Companies book)?
As said, you literally need no further models for the very first game(s). (How many games depends on reinforcements and promotions rolls.) For Rohan, you will want a blister of Royal Guard on foot quite soon after. Some mounted ones and an outrider or two later - they're currently unavailable from GW
as they are probably being repackaged for the upcoming Rohan supplement. You can grab some additional hero sculpts to use for your custom minor characters, but you don't need them. For the Army of the Dead, I guess you just want a couple of the mounted figures (again, not sure of that company, but the range doesn't contain much else). There are also some mercenaries you can get, but you can look at those once you have the book. One thing worth noting is that the BC
book does not contain profiles, as the range of warbands basically contains all the troops in the game. You don't need the Profiles books (for Lord of the Rings, or Hobbit for Gundabad if doing the proxy thing) immediately though, you can get the profiles from Lonely Knight's excellent armybuilder: http://lonelyknight.0fees.net/?i=1