OptiPlex 960 Upgrades

Solution
Here are the builds for that.
https://www.userbenchmark.com/System/Dell-OptiPlex-960/527
The best setup would be a Q9650, but the Q9550 is cheaper if that's an issue. 8 GB DDR2-800 low density RAM.
A ZOTAC GTX1050 2GB Mini will fit with no mods, and run off of the Optiplex PSU. ZOTAC GTX 1050Ti 4GB mini also fits but in a very few apps. it would like more than 8GB RAM.
With a PSU upgrade an MSI GTX1060 3GB OCV1 can be made to fit by cutting the bracket in 1/2 and trimming the plastic at the rear back to the end of the card. It's perfectly happy with 8GB RAM.
Other 1060s will have an DVI port in the 2nd slot that will hit the PS2 ports. The ZOTAC cards are 2 slots wide but have a 1 slot bracket. Other cards won't fit.
CPU cooler...
Streaming and gaming on a 10-year old PC with a Core 2 Duo and a 255 watt PSU will be very challenging and it could be more expensive than getting a new low end PC... unless of course you search for used components.

Which CPU does it have and how much RAM?

 
Here are the builds for that.
https://www.userbenchmark.com/System/Dell-OptiPlex-960/527
The best setup would be a Q9650, but the Q9550 is cheaper if that's an issue. 8 GB DDR2-800 low density RAM.
A ZOTAC GTX1050 2GB Mini will fit with no mods, and run off of the Optiplex PSU. ZOTAC GTX 1050Ti 4GB mini also fits but in a very few apps. it would like more than 8GB RAM.
With a PSU upgrade an MSI GTX1060 3GB OCV1 can be made to fit by cutting the bracket in 1/2 and trimming the plastic at the rear back to the end of the card. It's perfectly happy with 8GB RAM.
Other 1060s will have an DVI port in the 2nd slot that will hit the PS2 ports. The ZOTAC cards are 2 slots wide but have a 1 slot bracket. Other cards won't fit.
CPU cooler listing are a little screwed up. When I look at the whole system images I see a plastic BTX cooling shroud that will take a T9303 heatsink. If yours has a black plastic cover with 6 holes in the top that's the answer.
When I search Optiplex 690 CPU cooler I get coolers for the LGA1366 XPS435T which I own, and know won't fit at all. Or metal enclosed cooler which I believe fit the Desktop size and USFF computers. They also have a unique bolt pattern, different from other Opti DT and SFF coolers.
 
Solution