Why can't "She is sad all the time" be an accepted answer? Isn't this what it literally means?
See it like "all the time" (hela tiden) means "always during a certain period", and "always" (alltid) means "always during an unlimited period, forever and ever".
So in Swedish 'she is sad' and 'she is sorry' are the same? if yes, are there specific situations you would use them in?
"Sorry" as an English adjective can mean "sad, poor, unfortunate" depending on circumstances.
Example useage in a joke: A: "I'm sorry." B: "Yes, you are. Now apologize."
I don't know. when I hover above ledsen, it says 'sad' and 'sorry'... :)
Yes, you can use "ledsen" to mean "sorry", and I would say it's mostly used in the context of apologizing. Saying "Hon är ledsen" in that sense sounds like you're apologising on her behalf. As for how you would know if it's ledsen sad or ledsen sorry, context.
Why do the adverbs sometimes come before the subject and sometimes come after? How do I know which ones work in each way?
There's a long post about word order that you can find from the sticky post under Swedish discussions, that might be helpful.
Disclaimer, not a native speaker here, but I do study Swedish outside of Duolingo. I don't have a super specific answer for you, but ibland will either start or end a sentence. It will not go in the middle. Alltid, aldrig, ofta, sällan, will come after the verb.
Sorry in an apologetic sense or in a bad state sense: "she felt sorry for breaking the vase" or "the house was a sorry sight to behold" ?
It can be either an emotional sense or an apologetic sense. Sorry in the bad state sense would be, at least in the context you provided, "sorglig". A sorry sight = En sorglig syn.
Formally as well, I'd say. I can't see anybody ever pronouncing the d under any circumstances, honestly.
Native here: My old school teacher actually gave me a mnemonic for exactly that. Alltid is alltid (always) spelled with two l's, aldrig is aldrig (never) spelled with two l's.