Will the MSI H110M PRO-VH fit into a Corsair Carbide 100r?
Yes. The Corsair Carbide 100R is a mid-tower case capable of fitting ATX, micro-ATX, or mini-ITX format motherboards. Your MSI H110M PRO-VH is a micro-ATX motherboard.
I'm planning to change my processor to an Intel Pentium G4560 since they say its like the i3 6100.
Although you can fit an Intel Pentium G4560 (7th-gen Kaby Lake CPU) on the H110M PRO-VH (released during the 6th-gen Skylake CPU), you *may* be required to update its BIOS in order to support 7th-gen CPUs. The H110 motherboards would definitely support 6th-gen CPUs out-of-the-box, but, for newer-gen CPUs, it depends. If the actual H100M PRO-VH motherboard you will purchase with a seller was manufactured PRIOR to the release of the Kaby Lake CPUs, then, you cannot boot it with that G4560. If, however, the actual motherboard you will buy was manufactured AFTER the release of the 7th-gen CPUs, then, there is a *chance* it will support the G4560 out-of-the-box.
It is highly recommended that you pair a 7th-gen CPU with a 7th-gen motherboard (such as the B250/B250M) to eliminate any issues on incompatibility/BIOS updates.
I'm also getting a graphics card, but I don't know what to choose from GTX 1060 3g/6g or Rx 470 4g?
The GTX 1060 is one of the top GPU's intended for 1080p/60Hz ultra settings gaming. The RX 470 is a bit lower in terms of performance but still capable of delivering the graphics power for 1080p-resolution gaming but might have to tune down some in-game eye-candy settings to get acceptable framerate, depending on the game.
Since the GTX 1060s (as well as th RX 480s) are the top-of-the-line GPUs for 1080p/60Hz, you *might* experience bottlenecking using a weaker CPUs such as the Pentiums or i3's. The CPU *might* (depending on the game, and your settings) hinder the maximum possible performance those GPUs could have provide.
The RX 470, being a slightly "weaker" card than the GTX 1060 / RX 480, would have "lesser chances" of bottlenecking by those mentioned CPUs.
What PSU should I use Corsair vs550w or Seasonic s12ii 520w + 80 bronze?
The Corsair VS series are one to the poorly-rated and unrealiable PSUs out there. The Seasonic, in contrast, is one of the best. Between those two choices, clearly, you should get the Seasonic S12II 520W. If you want modular cabling, you can also get the Seasonic M12II-520 EVO. If you have the budget, you can also opt for the more efficient Seasonic G-550 (550W), semi-modular.
Hope these answers help.