I like having company that acts as my tank, which Lydia does. I'm sure you know the mechanics, but I save before every big fight, and the only time Lydia has died was when I accidentally hit her with a spell. The spell does not kill her outright, but the mob beating on her does kill her. No fight I've faced so far was so difficult that I could not stand a re-do. Recently I reset her specs and she holds up better in fights.. It is silly, but I like having a companion and I am attached to her.
Maybe you're right! I actually am starting to feel guilty having her hang around home doing nothing instead of out adventuring.

Ok, that's it. When I play tonight, Lydia & I are going out on a battle-date!
p.s. How do you reset her specs?
Being arrested for walking through an open door is strange and must be a glitch. I've not had that happen to me, although I have been arrested for picking a lock to enter a residence. Usually my intent is to roust someone out of bed to talk to them about a quest.

My impression is that in Skyrim a guard must see you, but I'm not sure about that. In Oblivion, I used a "no psychic guards" mod because it seemed no matter what if you lock picked a residence, a guard appeared out of no where to arrest you.
I thought so too - I didn't think you could just wander into a 'private area', but in Understone Keep, that's what happened to me. Since then, I've become Thain (or whatever the title is) and can go anywhere unhindered. Still weird..
AFAIK, regarding the guards seeing you, if you're sneaking and "Hidden", you're ok.
Why on Earth would they arrest you for entering the Shrine of Talos? I wonder are you aligned with the Stormcloaks?
Don't want to post spoilers, but there's a certain quest that triggers this. However, even once you complete the quest, every single time you return to the Shrine, the same event plays out again.
When someone hands you a quest and says it's marked on the map, the best thing would be to untrack all quests except the one you just got and see if one of those marker arrows appears on the map?
Often, there is no quest. For instance, if you talk to a mercenary, and they "give" you one of their jobs; they simply mention there's something to be done nearby, but there's no quest/objective, and the name of the location doesn't appear on-screen.
On another note - I wonder what would happen if you tried to 'mess up' the quests by killing a main character, such as Ulfric Stormcloak, early in the game. Is he invincible? Or very hard to kill? Or do lots of quests become unavailable?
One more annoyance too - there's one quest where you're trying to get someone out of a prison. I was able to sneak into the fort in question, but even though I wasn't seen, the quest immediately failed once I entered. Eh? How? Seems a bit random.