Unless that cooler is some actual brand name add-on cooler bundled with that CPU (with a normal retail value $30+) you cannot use it. Almost all CPU's are sold with a shitty little CPU cooler, some tiny fan connected to the worlds smallest heatsink just so the chip could potentially operate if installed on a motherboard right out of the box. You always want to invest at least $30 on an after-market cooler with a decent sized fan and fair sized heatsink, and then apply a name brand thermal paste (like Arctic Silver) as opposed to any included paste that may come with the CPU or the Heatsink/Fan.
This is a nul argument here. We do not need to worry about any of this, particularly with regards to PR. ANY modern Desktop CPU of the Intel persuasion, with multiple cores, will kick the ass of any old game like PR even though, admittedly it could not take advantage of the multi-core capabilities of the CPU. A single core off a modern multi-core PC can handle ANY game from 2004 (mods/updates irrelevant), unless it's the lowest end model of a CPU that is made (like some entry level mobility CPU).
Get a modern Intel multi-core CPU, the most expensive you can afford. The GPU (graphics card) you buy should cost about the same. The MoBo may cost a little as a third to a half of the CPU/GPU. These factors dictate which level of performance you can afford with your budget. Drop your budget onto the CPU/GPU/MoBo minus the cost of RAM. So, you say $500 ... spend $200 on a CPU, spend $200 on a GPU, spend $100 on a MoBo... you'll need another $60+ for a set of RAM, 2x4GB is enough to get you started because:
RAM is so nice, you can buy it over time... Start with the entry level you can afford, usually 60-100 for a pair of 4GB RAM for a total of 8GB, and later you could do that again when you can afford it (for a new total of 16GB, more than enough for modern multi-tasking such as streaming and also recording while gaming while watching some YouTube and also leaving 5 or 6 other programs minimized cuz you can't be arsed to close them)
If I were you, I'd try to budget more like $700-$750, because around those price points, on that site you linked, it seems the good GPU's and CPU's are around $220-250 (converted) and even up to $300, meaning by our example, you'd spend ~$250 on CPU, ~$250 on GPU and ~$120 on a MoBo