flashed motherboard atx

Solution
Yes, a GT1030 can run CSGO pretty well, but mostly because it's more CPU-dependent than anything to do with the 1030.

The A6-7400K isn't a strong CPU to begin with, with the benefits (mostly) being the onboard GPU.
By using a discreet GPU (assuming you've connected the display to the GPU and not to the motherboard), all you're doing is not using the onboard GPU.
It doesn't miraculously improve the CPU performance, unfortunately.

What resolution/settings are you playing at? Are you monitoring your CPU/GPU/RAM utilization?
If not, something like HWMonitor can do that for you: https://www.cpuid.com/softwares/hwmonitor.html
I assume you're talking about flashing the BIOS? No, there's no impact (good or bad, generally) to gaming performance directly.

Depends what the BIOS revision is for. It may improve compatibility for RAM speeds etc, which may have an indirect impact on gaming performance.
 
Yes, a GT1030 can run CSGO pretty well, but mostly because it's more CPU-dependent than anything to do with the 1030.

The A6-7400K isn't a strong CPU to begin with, with the benefits (mostly) being the onboard GPU.
By using a discreet GPU (assuming you've connected the display to the GPU and not to the motherboard), all you're doing is not using the onboard GPU.
It doesn't miraculously improve the CPU performance, unfortunately.

What resolution/settings are you playing at? Are you monitoring your CPU/GPU/RAM utilization?
If not, something like HWMonitor can do that for you: https://www.cpuid.com/softwares/hwmonitor.html
 
Solution
While dated, the i7-2600 is a great CPU. Consider the new route though, and factor that in to any pricing consideration you look at.

A new Ryzen3 1200 (while the budget offering, is a strong performer), B350 board to allow you to OC a little + DDR4 costs right around $240

PCPartPicker part list / Price breakdown by merchant

CPU: AMD - Ryzen 3 1200 3.1GHz Quad-Core Processor ($109.86 @ OutletPC)
Motherboard: ASRock - AB350M Micro ATX AM4 Motherboard ($59.99 @ Newegg)
Memory: Patriot - Viper Elite 8GB (2 x 4GB) DDR4-2666 Memory ($72.99 @ Newegg)
Total: $242.84
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-09-25 17:42 EDT-0400

To invest in an older platform, which you'll struggle to find a motherboard replacement for if the used one you buy fails, it has to make sense.
To me, I wouldn't look to spend more than $100 on an i7-2600 + compatible motherboard.
 
New? Not a lot in all honesty. I'd look to get by with what you have for now, until you have a minimum $250 for your upgrade.

Used? You might be able to find a system with an i5 or i7 (even 1st Gen), ideally a minimum of 2nd Gen IMO for $100. But it depends on your location.