VicioGamer47

RX 460 is ~10% slower than the 1024 shader RX 560, but there are RX 560s that have the same shader count as the RX 460 (896).  It is hard to tell which is which sometimes as the packaging tends to not display the shader count.
 
Anyway... If you are going to their store in person and they are trying to charge the same amount for the in stock RX 460, try to haggle them down by at least 10%.  And remember, the same limitations apply, has to be low profile with no external power connector.

Also...
 
DP to VGA generally works much better than HDMI to VGA, as the standard was made with that type of conversion in mind.  HDMI to VGA is more of a hack/workaround and may present problems with resolutions available (wrong aspect ratio, unsupported resolutions, etc).

Incorrect.
 
I am now running an RX 560 on an old school BIOS motherboard, 1st gen socket AM3/DDR3.  The board is 100% UEFI unaware and the RX 560 does not care.  Most GPUs have both UEFI and BIOS firmwares smashed into one.  The code checks for BIOS or UEFI and loads the proper firmware at boot.

As I said, it would be best to go with the computer (just the unit, not the monitor) to that person and ask him to allow you to install the video card on the spot and check to see if you have some image.
 
These modern video cards work only in systems that have BIOSes with UEFI extensions or that are UEFI aware.  UEFI is a sort of BIOS replacement, which was standardized around 2007 and majority of computers from around 2010 onwards had some kind of UEFI support in BIOS. But, big PC manufacturers like Dell and HP had series of computers that had BIOSes without UEFI support even for a few more years after UEFI was common everywhere.
Your motherboard's BIOS should be UEFI aware so the video card should work, but there's a slim possibility the BIOS doesn't have that UEFI stuff in it.
 
You should also consider updating the bios.
From the pictures, it looks like you have bios J01 version 2.15. If I got it right, the latest bios is 2.33A released in May 1st, 2019 : https://support.hp.com/us-en/drivers/selfservice/hp-compaq-6200-pro-small-form-factor-pc/5037900
Version 2.15 was released in Nov 28, 2011
since then you had lots of updates, 8 or 9 versions of the bios for that system.
 
 

Yes, DVI-D is DVI, Digital only so if the video card has a DVI port, then you can use a DVI-DVI cable.
If there's no DVI cable, you can use DVI-HDMI adapter and then use HDMI cable.
 
Double check that you actually have DVI connector in the back. In the past there used to be monitors with the same product model but some versions with S or V (for "value") at the end which had only the VGA connector.
 

No, DVI to VGA adapters will not work.
 
Those adapters simply connect wires between the VGA pins and some pins in the DVI side, which carry the analogue signal VGA needs.
However, modern video cards don't put that analogue signal on the DVI pins anymore, so those DVI-VGA adapters will no longer work.
 
A DVI to HDMI adapter works the same, but the wires make connection between DVI pins that have digital signal to corresponding HDMI pins.
So, if your monitor has a DVI connector but your video card doesn't have a DVI connector, you can use a DVI-HDMI adapter (which works both ways) along with a HDMI cable. You plug the DVI-HDMI adapter directly in the back of your monitor and then you can use a HDMI cable between the video card and the DVI-HDMI adapter.
The DVI-HDMI adapter looks like this (pics from the ebay links above)


 
These will NOT work :

 
If you absolutely need VGA, then you'll need to get HDMI to VGA converters, or DisplayPort to VGA converters  - these contain a chip inside them which takes the digital signal and produce an analogue signal that then gets put on the VGA side.
Here's example of such converter:
https://www.ebay.com/itm/HDMI-Male-to-VGA-Female-Video-Cable-Cord-Converter-Adapter-For-PC-DVD-HDTV-1080P/333315269145
https://www.ebay.com/itm/1080p-HDMI-to-VGA-Adapter-Cable-Converter-for-HDTV-PC-Monitor-Laptop/142921438721

More info on HPs website regarding what I assume is your computer.  Looks like an HP 6200 SFF from the specs at least.
 
https://h30434.www3.hp.com/t5/Desktops-Archive-Read-Only/Using-GTX-750-950-RX-460-with-HP-6200-Pro/td-p/5664119
 
HP says no way, users installed cards with no problems.
 
I have an RX 560 and a VERY old board that is pre UEFI, I can test to see if it works or not.  I am almost 100% sure it works.
 
Ill post back once I get to the office where that PC is and test.

Correct. However, the video card doesn't really HAVE TO stay inside the slot.
 
If you use your computer laid flat / horizontal, you can place the video card inside the computer somewhere (and make sure the back side doesn't touch anything - for example cut  a plastic rectangle from something and place the video card on the plastic rectangle so that nothing touches the insides) and then use a pci-e riser cable to connect the card to the actual slot. 
Here's an example of such a cable: https://www.ebay.com/itm/PCI-E-PCI-express-164pin-16x-16x-Flexible-Extender-Riser-Card-Flexible-Cable/113860936719
Here's another seller : https://www.ebay.com/itm/20CM-PCI-E-1X-4X-8X-16X-Express-Riser-Card-Extender-Extension-Adapter-Cable-R/173075373691
 
Also, you probably use your current monitor with a VGA cable. If so, the modern video cards no longer output VGA signals. so you'll have to use a DVI cable, or (if the video card no longer has DVI), you'll have to use a HDMI cable plus a HDMI to DVI adapter (these are cheap, around 1-2$)
ex.
https://www.ebay.com/itm/DVI-D-24-1-25-Pin-Male-To-HDMI-Female-Adapter-Connector-Gold-Plated-Stylish/253097203386?epid=2204507252
https://www.ebay.com/itm/DVI-D-24-1-25-Pin-Male-To-HDMI-Female-Adapter-Connector-Plated-Stylish-ok/142321337374
 
edit: also yes, these video cards require UEFI aware BIOS... but those HP 6200 should be modern enough to handle these cards.
 
It would be best to go with the computer to that guy and test the video card at his place.
I can understand 70$+ may be a lot of money for you, so ask the guy if you can check the video card at his place because you're not sure the computer supports it.

A random RX 560 will not fit, it has to be a low profile card.  If the 560 doesnt work the 1030 would also likely not work either.  Make sure to update your BIOS from HPs website, check for any compatibility notices.

The recommendations are high because the manufacturer of the video card doesn't want you to use no-name chinese power supplies that say 300-350w on the label but in reality can only do 150w and damage the card, and then have to send you another card because it may still be under warranty.
It's safer and cheaper for them to say 400w.
 
Your power supply is low wattage and a decent one with good efficiency, so it can do up to at least 200 watts on 12v just fine.
Your CPU probably uses around 40w, the motherboard with the ram and fans probably around 25w, the two hard drives another 15w ... so you're looking at peaks of around 80w for your system without a video card.
If you don't believe me, see this : https://www.legitreviews.com/intel-core-i3-2120-3-3ghz-sandy-bridge-processor-review_1650/16
It shows 65w for total system power under prime which abuses the cpu (that's cpu,mb,ram,fans and idle hdd)
 
That leaves you with around 150w for everything else you may have.
It would be fairly safe to go with a video card that uses up to around 120w.
 
Also as a sidenote any video card without a 6pin or 8 pin connector will use less than 75w - a video card is not allowed to take more than that from the slot.  Your power supply probably doesn't have a 6pin connector, but you can buy an adapter which converts a molex or sata connector to 6pin pci-e
 
You have these (first number is idle/windows/browsing net, 2nd is in games):
 
RX560 = 9W/79W
RX550 = 7W/45W
RX460 = 13W/60-80W
 
GTX 1660Ti = 9W/125W (peak 135W)
GTX 1660 = 8W/125W (peak 135W)
GTX 1650 = 8W/80W
GTX1060 = 6W/122W
GTX1050Ti = 6W/75W
GTX1050 = 6W/75W
GT 1030 = ~ 30-40W
GTX960 = 7W/118W
GTX950 = 8W/103W
GTX 750Ti = 4W/57W
 

It pulls ~60w.  Combine that with your CPU at 65w and you are only at ~125w.  Add in everything else and you might be at 140w, well within the power limits of the PSU.

You can't game on a GT710. It's barely better than the onboard video. Can't imagine you can stream much either on a 2011 dual core.
Also, the GT710 isn't a $80 card, it's a $20 card.
 
Either way, to answer your question better, the better value option would be a low profile GT1030, GDDR5 version obviously.
There is a chance however that your motherboard will not accept it, as HP is known not to support everything that physically fits.
 
Update the BIOS first to the latest available version before installing a new graphics card.

i dont see how adding a gt710 helps. 
 
you could probably get a cx550 + a hd 7850 for 80$ total. 
 
it wouldnt fit, and making it work it you would need some sketch, but honestly. i think you are willing to give up some looks. 

GT 1030 half height.
https://www.newegg.com/msi-geforce-gt-1030-gt-1030-2g-lp-oc/p/N82E16814137371
 
Or RX 560 half height.
https://www.newegg.com/msi-radeon-rx-560-rx-560-4gt-lp-oc/p/N82E16814137425R?Item=N82E16814137425R
 
RX 560 is the better of the two.
 
There are half height GTX 1050s and 1050 Tis, but they are generally way overpriced.