I like that in Swedish "to feel" can also mean "to know"! Getting philosophical here but sometimes feelings can be the best thing to follow :)
Hmm? Spanish has a separate verb, sentir, which means "to feel (an object)" (and sentirse, the reflexive form, to mean "to feel (a feeling)"). Conocer means "to know of" or "to be acquainted with". I don't believe there's any case in which conocer and sentir are interchangeable.
känner mig is for how you feel (using adjectives) – when you could say I feel happy or I am happy – that would be jag känner mig glad. You're describing yourself, so you need the reflexive form.
känner is for feelings you feel (using nouns) – when you could say things like I experience happiness or I am overwhelmed by happiness or whatever – in those cases you are describing a feeling that you are experiencing, not yourself, so you don't use the reflexive form of the verb.
"At peace" means that your are calm, and the state of your mind is peaceful. You are not agitated or angry. If you meditate, I think this is the way you would want to feel.
"At " or "at " is not a general pattern in English. For example, all of the following are INCORRECT: "at angry", "at anger", "at happy", "at happiness". It's best to think of "at peace" as a specific phrase and just learn what it means.