My assassin and companion characters don't go through the main quests. They follow the Dark Brotherhood, Companions, and radiant quests. In their adventures, they get to acquire dragon shout words on the wall. My question is, if the characters don't do the main quests, can they encounter dragons elsewhere and get dragon souls in order to activate the dargon shouts? Please cite example side quests where dragons can be encountered. Thanks for your help!

nope

You HAVE to kill the first dragon at the Western Watchtower to initialize dragons spawning. So no.

If you don't do Dragon Rising, you won't get any dragon shouts.

Ok I understand I have to do Dragon Rising, after that, will dragons begin to appear in side and radiant quests? Thanks!

fixed.

Yes.