Well, lets start with the build as it stands. You have a 2.83ghz Core 2 Quad - you probably have the Q9500 series which is an LGA 755 motherboard and was released circa 2010 (about 5-6 years ago).
R7-240 is a 'low end' GPU, so-so FPS rates at about 1080p gaming (30-40 max) at medium settings. No high settings - otherwise you'll drop down even lower. Even the older HD 7770 has double the FPS rates that the R7-240 has. It really is a step up from integrated graphics, but not by much.
Now, what would be causing lag? The CPU isn't bad, it scores about mid-5000 on geekbench - but it's half of what a good Haswell i5 is capable of and is actually about what an i3-3220 can do. That means it's still a 'decent' gaming CPU, but it's getting close to end of life. Upgrading would benefit in both usability, and would also seriously reduce heat and power usage.
The GPU on the other hand, is more likely the culprit. It really is a bottom of the barrel discrete GPU. An R7-260 series would have been MUCH more powerful, and probably a better overall buy.
As for the 32bit vs 64bit question - if you do wind up upgrading your system to something newer, you will definitely benefit from at least 8GB of RAM, which to use all of you would NEED 64bit Windows.
To be honest, I would:
Sell the parts on eBay. All of them.
Then get the following:
i3-4170 CPU ($100)
low cost LGA 1150 motherboard (Gigabyte makes nice ones - try the GA-B85M-HD3 - nice, compact M-ATX board. ($75/ebay)
2x4GB DDR3-1600 RAM (8gb total, aim for a name brand if you can) ($30/ebay)
R7-260X if you have a decent PSU OR a GTX-750ti if your PSU is of 'moderate' capabilities. (260x = $70, 750ti = $100 on ebay)
That would give you a much more effective gaming machine without breaking the bank - about $275.00 total.