IS A MAC PRO (MID 2010) WITH A ATI Radeon HD 5770 1024 MB VIDEO CARD COMPATIBLE WITH MOJAVE?

No. The 5770 does not support Metal.

There are only two official cards for a 5,1 Mac Pro that support Metal. That is, cards that were designed for these Macs. Not flashed PC cards.

AMD Sapphire HD 7950 for Mac

EVGA GeForce GTX 680 Mac Editition

The GeForce is much easier to find, and roughly half the price the Sapphire goes for.

Run OpenGL Extensions Viewer

https://itunes.apple.com/app/opengl-extensions-viewer/id444052073

The Summary page will show if your graphics card supports Metal, which is needed for Mojave.

Apple have been very unhelpful regarding detailing what video cards in a MacPro5,1 i.e. 2010 and 2012 are suitable for Mojave.

Firstly your current card is definitely not suitable.

As Kurt suggests The GTX-680 and the Radeon HD 7950 are by common consensus considered suitable as long as they have Mac firmware on the cards.

Apple more recently did provide some additional advice, see - Get Ready for macOS Mojave - Official Apple Support

Unfortunately it is still confusing. Neither of the cards listed, the MSI Gaming Radeon RX 560 and Sapphire Radeon PULSE RX 580 have ever come with Mac firmware on them. This would normally be a problem and at a minimum would mean you could not use FileVault or RecoveryHD.

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[Edited by Host]

Actually Recovery HD can be booted with RX 560 or 580. It loads the OS drivers and is not dependent on the EFI driver. 🙂