Which BM pcie cards will downconvert 4k to 1080?

Will the mini monitor card (standard or 4k) down convert or do I need to go with a decklink extreme card to avoid constant signal path changes?

You rather need bit more expensive model, like Decklink Pro (and up).
All cheeper ones do only HD to SD downconversion.

You rather need bit more expensive model, like Decklink Pro (and up).

and this will also depend on your version of DesktopVideo.

I use an outboard AJA 4K2HD converter.

jPo, CSI