How do I get to the mirror world on battlefield? And what are the rewards for staying per 10 hours on the portal? It's written in the portal's description in the battllefield but it seems very complicated to me.

Before you can place your squad on the Mirror world, it will take some time to "research" how to get there. This happens automatically when you press the magnifying glass icon, and will take between a few hours to several days depending on your squad 's research potential. You can increase this potential and research faster by completing various tasks with each pet. 

I might not be the best person to answer your second question, but given the description it seems that you get 1000 silver for every previous tactic win your squad has gained, plus an extra 4000 silver, plus a symurgh or throdama. You can choose which one you get once you get to the portal space. 

I see. I've never thought the research potential of a pet ever mattered until now. Thank you for clarifying.