Frida from Dawn star has a radiant quest. For some reason she's not giving it to me. Any pre-requisite I should do before I can actually get it?

Frida's quest is one of several possible http://www.uesp.net/wiki/Skyrim:Dungeon_Delving_(Caves) radiant quests given by different NPCs. Check the list at the link, if you have any of those other quests already active in your journal, you can't get a new one until you finish the one you've already got.

My biggest problem with Frida is that (in my games at least) she sometimes refuses to open her shop for days at a time. Sometimes she'll actually be IN the shop, but it's locked and she treats me like a trespasser if I console command (or lockpick) my way in there. Other times she stays in the local inn all during the day when she should be at the store. If she's in the inn I've found that going there and talking to her will sometimes kickstart her stuttering AI and make her go back to the shop and pick up her shopkeeping schedule again.

Protip: if you get Frida's quest and it sends you to Forsaken Cave (50/50 chance as there are only two possible target caves), go to Windhelm and get the quest from the alchemist there as well. It will always send you to Forsaken Cave and that way you can do two quests on one trip.