Dell PowerEdge R720 not recognizing RAID card

Yes, in the general case you can install PCI-E cards in any PCI-E slot that will physically accept the card. If there are fewer PCI-E lanes available than the card requires, it should run a little slower but still run. If there are more lanes, it will just use what it needs.

PCI-E 1 and PCI-E 2 should not be a problem either.

Regarding Windows 7 not finding the RAID controller, I do of course assume that you're using the proper driver for the RAID card? And that if you're installing the OS onto a RAID volume, that you're loading the drivers during installation...

If all this is the case, then my guess is that you're using a consumer class desktop motherboard. Some desktop boards have PCI-E x16 slot which will only work with graphics cards. This goes against the intentions of PCI-E, but some manufacturers don't bother to do anything about this, since desktop machines almost always use the fastest PCI-E slots for graphics. Try contacting your motherboard maker / PC suppliers support..

+1 to all of Evan Anderson's answer (though I'm not familiar with opROM execution problems).

You may also want to investigate the specifications of your server to find how many PCIe cards of each type are supported, and whether there is any "slot priority" that you should be following (these factors exist even with Dell servers). Just because you have a PCIe slot for the card doesn't mean it's getting the necessary power and bandwidth to function properly.

As a work-around (if you can't figure out how to get the PERC card working) you may want to consider using a SAS HBA (e.g. Dell's SAS 6/E card). This would be presenting each individual hard drive directly to the OS though, which would require something akin to software RAID (like Storage Spaces in Server 2012, which is actually validated for use with that type of setup).

Edit 2014-09-11

Any advice on where to obtain the H800 drivers for Windows 2012? (Or equivalent LSI drivers?) I can only find Windows 2008 R2 64-bit PERC drivers so far.

The Server 2012 drivers for the PERC H800 card are provided by Microsoft via Windows Update. If you can get far enough to see the card listed in device manager, "Update Driver Software..." should handle the rest almost flawlessly.

I've looked in the BIOS previously regarding PCI-e settings, and there was no signs of slot priority that I recall. Also, this is the first expansion card we've added to the server - so regardless of any slot priority, I'd think it would allow at least one card without modification.

Slot priority (if any) is typically determined by the server manufacturer, and is not normally a configurable option in the BIOS settings. You should reference the documentation of the server (e.g. an installation, administration, or user guide) for details on which slots should be used first for specific types of cards. It could be that the slot you're using won't support a single external-storage card alone, even if no others are present.

Try testing in alternate slots with the PERC card. If that still doesn't work for you, look into purchasing a SAS6/E HBA or the LSI 9207-8e - I believe that either of these are supported for the MD12xx enclosures for use with Storage Spaces (I've tested both in lab with no problems, but cannot find supporting documentation for the MD12xx series enclosures at the moment).