the lobster you speak of is a cray fish , the reason he was eating the dead fish is his job of keeping the tank clean , think of him as a janitor , the fish was most likely sick and died.
yes you can keep him with fish he wouldn't be able to catch healthy fish as they would be to fast for him. be sure you have a tight lid on your tank as they are great escape artists. be sure to keep a rock cave for him to hide as they enjoy exploring as I found out.
here is a fun little story I can tell about these little guys I encountered
I had 3 a little over 20 years ago as my kids wanted them , one day I went over to feed to find one missing looked and looked but couldn't find it.
next day I went to feed again a second one was gone so now I'm looking at the cat.
another day passed the last one was still in the tank . sitting in my chair I saw the very last one climb out. I decided to see were he goes as the other 2 already made the trail.
a hour passed he was 1/2 way between his tank and the kitchen , than he finally made it into the kitchen , he darted across the floor and under the fridge he went , so I pulled out the fridge to find all 3 sitting in the condensation drain tray staring at me like what you want,
these little buggers found a new water source and thought they would have a new home
I was amazed on how the cat didn't catch them making a meal . any way I snatched them up and put them back into there tank. I wound up closing off any open areas around the filter heater ect so they couldn't get out again , needless to say we had them about 5 yeas after that little incident ,
my kids got bored of them so I gave them to a friend for his kids a week went by I got a call all 3 are gone , I giggled than I said check under your fridge sure enough they were there ,