Regarding Animal Farm, did anyone follow Stalin's ideas after his death, and when Napoleon dies will a new leader will be elected similar to him?
When Napoleon dies, will a new leader be elected with similar ideas?
I guess this depends on what you mean by "Stalin's ideas." In a way, people did continue on with his ideas for almost 40 years after he died. The Soviet Union continued to be communist and the members of the Party continued to have all sorts of privileges that regular citizens did not.
On the other hand, the worst of Stalin's ideas did not survive. The guy who ruled after him, Nikita Khrushchev, denounced Stalin's way of ruling and did not follow it. There were no more mass purges and mass executions after Stalin died, for example. So the basic idea of communism survived, but Stalin's brutal ways did not survive as much. (There was still a great deal of repression, but not nearly as much as in Stalin's time.)
So if Animal Farm follows real life, the next one to take office will surely be a pig who believes in Napoleon's general ideas. But this pig will presumably not do things like killing all those animals for supposedly helping Snowball.
I will take the second question. Part of what makes Napoleon's reign on the farm so horrifying is that it extends over a period of time, where those who struggled have been wiped out by memory. The animals who now remain on the farm are not the ones who understood the basics of the revolution, nor are the ones who are going to spark social and political change. Instead, they are young and because Napoleon's rule has been present for so long, his rules are all they know. This ensures that Napoleon's reign will continue, even after his death, for he has consolidated his power so thoroughly, with an internal apparatus that is so driven to maintain his power that nothing will stand in its way. There will not need to be an election after his death, as some new "pig" will take over his position. It is this irony, the one that sees animals and humans as no different from one another, that ensures that Napoleon's reign will continue even after he is gone.