Hello Ksenia Z.,
Geist in gernal refers to your inner life in contrast to your body. A destinction between Seele and Geist is somewhat vague. In direct contrast to each other Geist usually focuses on (active) thinking, intellect and rationality and is associated with the brain, whereas Seele comprises your feelings and longings and is associated with the heart. For your examples:
1. Im Geist getrennt - you have different oppinions or different ways of thinking and are probably constantly arguing. Im Herz vereint - you agree on the most fundamental level, you have a common goal and long for the same thing (or each other). Geist = Denken, Verstand
2. Verstand/Sinn would work here, but i think its a bit too restricted. Depending a bit on the context Geist might include additionlly Seele, and should be more seen as inner life versus body (represented by Lunge).
3. Verstand works perfectly here, as explained above.
Greetings, Jens.