I asked on a SubReddit about recommendations for a new build, but didn't get an answer. I am posting it here as it includes the same information you require. I plan to purchase the components from different places and assemble myself.
What will you be doing with this PC? Be as specific as possible, and include specific games or programs you will be using.
The most intensive tasks are numerical computations and gaming. A game I would play is PlanetSide2, which requires a stronger CPU than normal for gaming, but this is fine since I need one for the computations.
What is your maximum budget before rebates/shipping/taxes?
600-800 USD (2200-3000 NIS). Whatever best performance/price is standing at in this range.
When do you plan on building/buying the PC? Note: beyond a week or two from today means any build you receive will be out of date when you want to buy.
I am a patient buyer who can wait for good deals, so I plan to finish buying the last component in the next few months. I understand that by then the recommendations will be slightly outdated. I suspect that the GPU prices will drop when NVidia ships the new Pascal series, so that will be a good time to get one.
What, exactly, do you need included in the budget? (Tower/OS/monitor/keyboard/mouse/etc\)
Everything which is not peripherals. CPU, GPU, motherboard, RAM, storage (HDD/SSD), PSU, fans, case.
Which country (and state/province) will you be purchasing the parts in? If you're in US, do you have access to a Microcenter location?
Israel. Here is a local store with reasonable prices locally: http://en.ksp.co.il (prices are in ₪, which is NIS. Google will convert NIS to USD easily). If ordering from ebay is cheaper (after shipping+import) I go for that.
If reusing any parts (including monitor(s)/keyboard/mouse/etc), what parts will you be reusing? Brands and models are appreciated.
Will buy peripherals (mouse, keyboard, monitor) separately. Headphones are ATH m30x, I assume the built in sound card will not make them sound bad.
Will you be overclocking? If yes, are you interested in overclocking right away, or down the line? CPU and/or GPU?
Don't mind overclocking if the build can handle it and the performance increase is worth it.
Are there any specific features or items you want/need in the build? (ex: SSD, large amount of storage or a RAID setup, CUDA or OpenCL support, etc)
I think that a small SSD for the OS and time-critical access is worth it.
For the HDD 1/2 tera (500GB) is enough, but if for a small price increase I get a lot of space increase then I'm fine with more.
Do you have any specific case preferences (Size like ITX/microATX/mid-tower/full-tower, styles, colors, window or not, LED lighting, etc), or a particular color theme preference for the components?
No, the computer can look like a clown as far as I'm concerned.
Do you need a copy of Windows included in the budget? If you do need one included, do you have a preference?
No, I have a copy.
Extra info or particulars:
I don't need a super-cool and super-silent PC, but please don't recommend me one that sounds like a drill or heats the room like an oven.
Prices here can be very different from the US/UK, so if possible, mention a few alternatives for components which perform about the same because they can be priced very differently.
I'm somewhat flexible in price and performance as I'm not aiming for a specific goal ("I want 60 fps in this game no matter what!"). I'm looking for the point between "add a few more bucks to get a much better component" and "adding a few more bucks won't get you a much better component".
Happy to answer questions and get tips.