The hard drive cables interfere with the GPU. It's fixable, you'll just have to buy right-angle cables for your drives. As far as the card itself, yes, a 290X should work with an Opti -010 generation's BIOS. Emphasis on should. Dell does some really crazy shit with its BIOS and AMD cards can and will have issues with Optiplex boards including the -010 generation, but I seem to recall those mostly being either very new cards (RX 480, 580, etc.) or very old, low-end cards. Still, Google things like "R9 280X Optiplex 9010" and look for people who are successfully running that setup.
The bigger issue is heat. If you have an open-air cooled 290X, I wouldn't bother. It'll spend its whole life throttling itself and everything else in that system with the amount of heat it puts out. If you have a 290X blower, eh, maybe. The rest of your system should be fine, but Optiplexes don't necessarily have a ton of airflow going on, and your 290X would be fairly starved. As a result, it would run hot and loud. Emphasis on loud. If you're ok running the Optiplex with the side panel off, then all of this is moot and you're fine, but a blower will still be loud AF because they just are.
So basically, if your BIOS supports a 290X and you have a blower, I'd use it. If you have an open-air 290X, I'd only do it if you plan to run with the side panel off. And yeah, you would need that 750W PSU. The good news is that the -010 Optis are the last that used standard PSUs and standard 24-pin connectors.