There is no quad channel on socket LGA775, only dual Channel.
The amount of boards supporting DDR3 are pretty limited. It was the time of DDR2 after all, and DDR3 gives close to no performance benefits over DDR2 on this architecture. And DDR2 was cheaper than DDR3 back when these boards were released, so demand for DDR3 boards was pretty limited.
The boards who support DDR3 are usually equipped with 4 slots for DDR2 (limited to 16 GB total) and 2 slots for DDR3 (limited to 8 GB total) which can not be used simulteanously.
The P45 and X48 chipsets generally support up to 16 GB DDR2 but the 4GB DDR2 sticks sold today are generally only usable on AMDs AM2/AM2+ platform. It has complicated technical reasons.
For 4x 2 GB of decent DDR2-800+ sticks you will face atleast 50€/$
Same goes for a motherboard, decent boards won't sell for below 45-50€/$ You will not the only one with interest in keeping their older system alive.
All these prices are actually for used parts!
In short:
■NO, you will not be able to run 16 GB DDR3 on a Core 2 Quad platform.
■You won't even be able to run 16 GB DDR2 on a Core 2 Quad platform.
■You would need atleast face an investment of $100 when staying with 8 GB of either.
■Both CS:GO and Overwatch will run fine with 8 GB, and have no issues even with just 4 GB (personal experience with such a system)
■Fast Core 2 Quads (Q9550/9650/QX9770) will sell for good prices (expect $60-$80+). So it might actually cheeper selling the CPU and investing in a modern platform with similar performance and the ability to run 16 GB DDR3.