Research before project is always a good idea. Upgrading HP processors is a project..... here would have been my advice:
1. Contact this forum and ask if it could be done to see if anyone else actually has done that.
2. Look up the latest Quickspec for your HP Elite Desk 800 G2 SFF.... that has a list of specific processors for your workstation that HP has approved for use for compatibility. The way unauthorized processors won't work is to have a white list of approved processors embedded in the microcode section of the BIOS. Sometimes later processors are added for a workstation because HP has tested them to be fully compatible, and you can find those by finding the latest QuickSpecs version. HERE is a quite recent one, version 31 from 10/16/17 for your box:
http://www8.hp.com/h20195/v2/GetPDF.aspx/c04818335.pdf (you can search to see if there is a later one)
3. Go to page 9 and see that the Intel Core i7-6700 6th generation processor is the latest approved for the SFF variant of your workstation (not the processor you mention). That is a 65W max TDP processor and so is your i5-6500 shown on page 10 so there are no cooling issues by the change.
4. Go to the Intel ark site via a google search for i7-6700, and on the ordering page there you'll see it has a single sSpec code of SR2L2 and retails for about 312.00 USD. Use that sSpec code to search eBay for a used one that is not bad from a reputable US seller..... recent used processors don't get a huge discount on eBay.
The latest microcode will be included with the latest BIOS update. If HP had added the 7th gen processor you bought you'd see that in the latest QuickSpec document released for your G2. The G3 version of your line of workstations will run that processor.
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