The Core i3 is a low-end CPU - though it'll run on any Kaby Lake mainboard you want, putting that on a gaming board is a bit like transplanting a Hyundai Accent's engine into a Porsche.
If you're looking for a high-performance/cost ratio, hold on until April 11. AMD's Ryzen 5 CPU is about to be released, and it'll shake up the mid-range (you'll be able to buy upper-mid range CPU for i3 dollars).
No. AMD is a competitor to Intel - the CPUs are different.
You've posted in so many places it's not clear what you're looking to do. If you are looking to build a system to replace your Inspiron 620s for gaming purposes, guidelines:
CPU should be no lower than an Intel i5 or AMD Ryzen 5.
RAM minimum 8 G
As much video card as you can afford - mid-range ones start in the $150-200 range
Mainboard as you wish - as long as it supports your CPU and intended use.
Case and power supply as you need for the board you buy.
Figure on spending about $600-700 for an AMD system configured for gaming, or $750-900 for Intel at the level you're looking for.
An i3 CPU is fine for office work but won't support even moderate gaming.