Well, being a doc myself and all, i hope to answer your questions a bit...
Soloing is difficult, without a doubt. Unless you are heavily twinked, which i do not assume as a fr00b and not someone's alt, you may be able to heal yourself nicely, but doing damage on your own and killing stuff takes lots of time. So it makes soloing slow (if not impossible), as you simply don't do enough damage usually.
The doc is probably still the most favourite class if a group needs a real healer. Unfortunately, in a "good" group, where people have tons of HP and deal large amounts of damage, while having high ACs and thus don't need the "full healing power" of a doc too bad, they do often stick to other classes, like the ones you mentioned. Especially the Keeper, with their healing aura that goes all alone, is sometimes enough already. Otherwise docs are still welcome in teams tho, i mean who got the "perfect unbeatable team" always, which are so good that they don't need a good healer?

Especially on raids are docs welcome, but thats no use for leveling and the general gameplay i'd say.
The oppinions on the PvP question are just split. Some swear docs at level 220 are just unbeatable in PvP. Some say they are unbeatable at lower levels. Being level 210 myself, i do know i don't have a chance against almost all classes even lower level, because they simply deal so much damage with their special attacks, perk actions, whatnot, that i often do not even have time to finish casting a debuff on them before i am dead already
But, apart from that, i don't really regret playing that profession. Just sucks sometimes, that your "main" isn't an unbeatable solo-kill-all-on-my-own-and-rush-missions-in-2-minutes-kill0r-class character, but hey... can't have all, right?
