You obviously can, since you did. ;)
Overall, you can play pretty much whatever you want. There won't be any "impossible" combination. Of course, there are combinations with stacking advantages (like Khajiit Nightblade), but overall you won't fail just due to some specific character combination.
Also remember that group dungeons are meant to be challenging, even if you're right at a dungeon's level. They're called group dungeons for a reason.
In PvP it's pretty similar. You may die often or you might not die at all, which will also depend on your enemies. There's no definitive answer to that, other than you caan play anything and compete with others. But don't expect to beat everything they'll be throwing at you, not even with some "perfect" combination.
And just to add for your last remark: Why would a templar with staff and heavy armor be something preferred? While a staff might be typical for healers, a templar with shield would be your classic paladin. In a similar way, you don't necessarily wield heavy armor. Most games feature healers in light or (at most) medium armor - exactly what you've picked.
Of course you won't have as much defense picking medium armor for example, but at the same time you've got more stamina to outsprint enemies, dodge roll more often, or just use stamina based abilities more often. In a similar way a light armor wearing templar would be able to heal more often. You'll always have to make sacrifices for gaining some advantage.