Unless it was possible in the incredibly early days (60s era), no - it's only ever been visible from a distance unfortunately. The hippo feeding (and riding) took place on the near side of the lake, as is visible from old footage. The drive through safari route runs directly next to the hippo, sea lion and gorilla housing so I don't think it's impossible this will become possible in the near future.
Nothing that hasn't already been technically answered by the links to planning permission and news stories above, but categorically - the entire old house has been demolished, and a new indoor complex has been constructed. It's been in the local news here in Bath recently due to local businesses being sourced for contractor work and materials. Notably this features a heated indoor pool which is a massive leap forward compared to the old house, but in fairness the Wiltshire winters didn't seem too harsh on the previous inhabitants. Aside from the lake itself and the main grazing paddock, the entire hippo area of the park has been completely redone. Quite literally nothing of the old house remains - it was demolished weeks before work started on the new one