I've searched and didn't come up with a solution for this:

I have a custom node type. The standard search block gives results on an extra page with all the different node types shown as tabs. Also, my custom_nodes show up twice: as custom_nodes and within "Content" (albeit not rendered as "custom_nodes").

I'd like to have a search block that starts a search only within my custom_node_type and hides other content types on the result page.
For the standard search, I'd like to remove custom_nodes from the general "Content" results, or at least render them as custom_nodes.

Any ideas on this?