Will my Mac Pro (Mid 2010) with an upgraded Metal graphics card run Catalina? If not, are there any upgrades I can do to fix this?

Only macs from 2012 can run Catalina. Specifically only Mac Pros (Trash Can) from 2013 can run Catalina.

Read Section 1 in the link below.

How to upgrade to macOS Catalina – Apple Support

This is an official Apple website so any other avenues are not allowed to be discussed.

I use an Nvidia GTX 680 2GB, a second hand card that I bought 2 years ago.

The drivers for the GTX 680 are built in to the OS, later model Nvidia cards may not

work because of the ongoing feud between Apple and Nvidia so there may be no

driver support for later model cards.

Also look for Mac Edition cards or 'flashed for Mac' cards, these include the Apple EFI

which gives you the Apple boot screens and access to the Startup Manager.

PC, (Non Mac Edition or non flashed) cards are often cheaper but do not give you the boot

screens, the Mac boots directly to the login screen or Desktop whichever is set. You will not

get the Startup Manager when requested on startup. The cards will work perfectly well.

A very popular choice at the moment is the Sapphire Radeon RX580, it is also new as

opposed to second-hand. There are some places which will flash your RX580 for

you, which means you sending it to them and paying about $100 for the service.

Graphics cards can become very expensive so look for one that you know will fit your needs.

The more expensive cards with large RAM are expected to be used for HD video editing

or 3D modelling, so if you do not do those sort of tasks on your Mac there is no point

in spending hundreds of dollars/ pounds etc.

You will also need to get a card that has the correct out put for the displays you are

connecting to. The GTX 680 has two DVI, one HDMI and one Displayport.

It may be worth your while asking this question on the Mac Pro Community here

to see if you can gain any more information from fellow Cheese Graters.

https://discussions.apple.com/community/desktop_computers/mac_pro

Mojave will be effective as long as your Cheese Grater keeps working.

Apples support, i.e Security Updates etc for Mojave will probably run out this time next year.

Apple only keeps it last three OS's up to date with updates.

At the moment it is High Sierra, Mojave and Catalina.

Big Sur is due out soon, so High Sierra will drop off.

If or when Apple introduce a new OS next year then it will be that new OS, Big Sur and Catalina,

Mojave will drop off the update list, but you will still be able to use it.

Mojave is great OS. Catalina not so.

Why not run High Sierra on one drive and Mojave on another

and evaluate between the two.

Yep that is right, only you can make that call, but the graphics card is the most likely part of the MacPro to fail

so at some point you will need to get a new/ newer one.