"homomorphic encryption, which is not performant enough"
It is fast enough on a per viewer basis, and in a DHT downloading the database doesn't mean it was all encrypted w/ one key. Each user encrypts his data as needed, or common groups of users encrypt data for each other with each others keys.
"If I make any mistakes on the database security"
This is why encryption is the underpinning. Sure you can still leak your private key like you can leak an SSH key today.
"in client javascript"
Nobody would use a distributed network where this was the case. In many cases (i.e. MaidSafe) they are developing a browser plugin for client side to communicate with the backend.
"where does system processing that is independent of user activity (scheduled tasks, etc) happen?"
Many of these now-being-designed systems have a pay-for-computing concept. Granted several (not all, unless you want to be limited by a single-file-line blockchain forever) have to agree on the results. Give some computing for other computes and get some. As for "scheduled task" timing issues are inherently difficult for these systems and I don't expect the "system" to trigger a job but rather a user to trigger it. Introducing timing into these distributed networks can be hairy.
The real problem that needs to be tackled is a way for the common human to hold his private key in his memory or some other non-digitally-retrievable way.