How to tell if these components fit in the Corsair Crystal 460x?

You can tell by cross-checking the specs of your case (http://www.corsair.com/en-us/crystal-series-460x-rgb-compact-atx-mid-tower-case) with the specs of each component you have.

Motherboard Form (ATX, mATX, ITX): ATX - fits.
CPU Cooler Height (up to 170mm): 84mm - fits.
GPU Length (up to 370mm): 279mm (using MSI GTX 1070 Gaming X - fits.

You would need a regular philips screwdriver. All screws are included already. Other screws are thumbscrews or tool-less connection.

All of your components are not outdated.

HOWEVER, apart from the physical compatibility you have to take into account the parts' performance compatibility and your usage compatibility.

In your partslist above, there are several parts you would have to change for best price/performance:

1) CPU/MB combo. You selected an unlocked CPU ("K" version) which was designed for overclocking. However, you selected a motherboard with a B150 chipsey that cannot support overclocking. Furthermore, the CPU you selected is a 7th-gen Kaby Lake which will not run on a B150 (or H110/H170/Z170) mobo without updating its BIOS. You would need a 6th-gen Skylake CPU to be able to boot and update the mobo's BIOS. If you will get a 7th-gen CPU, it is best to pair it with a B250/H270/Z270 mobo as Kaby Lake CPUs will run out-of-the-box. If your CPU is unlocked (such as your chosen i7-7700K), get the Z270 mobo. If you opt for a locked CPU (those non-"K" versions), then get the B250 for better price/performance.

2) CPU Cooler. That low-profile CPU cooler is not the best out there, esp. if you chose an overclockable/unlocked CPU. Your case can fit tower-type heatsinks for better thermal dissipation.

3) GPU power and monitor. The GTX 1070 is ideal for 1440p/60Hz or 1080p/~120Hz gaming. If your monitor is just 1080p/60Hz, then a GTX 1060 or RX 480 would be enough to save money.

4) PSU is not the best out there although enough juice to power your rig. If you want better quality/warranty, I'd suggest the EVGA SuperNova G3 or G2 550W or the Corsair RM550x. If budget is limited, the Seasonic G-550 or Seasonic M12II-520 EVO would be a good option.

EDIT: I just saw the SSD you chose. The Samsung 750 EVO (I thought it was the 850) is one component that is outdated. Better to get the most recent model.