First, understand that not all layer-2 protocols are broadcast protocols.
I'm assuming that in most cases the L3 broadcast is also encapsulated
into an L2 broadcast frame, but is this standardized?
When you send a layer-3 packet, the layer-2 frame needs to be addressed, and the layer-3 address must be translated to the layer-2 address for the layer-2 frame. That is the purpose of ARP for IPv4. A layer-3 broadcast address will be translated to the layer-2 broadcast address for the layer-2 frame. (It is possible for someone to craft a frame with a layer-3 broadcast packet and not a layer-2 broadcast address, as I have seen on Stack Overflow, but that is really pointless.)
For a limited broadcast, does the sender always encapsulate into an L2
broadcast frame? Are there cases when a limited broadcast would not be
sent to the L2 broadcast?
When the layer-3 destination broadcast address is translated to the destination layer-2 address, it will end up as the layer-2 broadcast address.
For a directed broadcast to an adjacent network, would the frame be
sent to the gateway MAC and the router then encapsulates into a L2
broadcast on the far end?
Remember that a directed broadcast is not known to be a broadcast on the source network. To devices on the source network, it is simply another address on a different network. The devices on the source network have no idea of the mask on the destination network, so they have no idea that it is a broadcast. The destination router that has the mask for the destination network does know that it is a broadcast on the destination network, so the destination layer-3 broadcast address will be translated to the destination layer-2 broadcast address.