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If you remove the hdd bay cage, then you would have enough room to install a full length gpu in the 7010 MT case. your hdd or ssd can be mounted in external 5.25 odd bay via an adapter. HD5870 card is 282 mm long. The case is 417 mm deep.
you can easily upgrade the stock psu to a 450-500 W psu to support this card. Dell sells many 460W oem psu for XPS and Aurora line with 2x 6 pin gpu connectors. used ones are cheap too on line. the size is standard for MT. suggested psu for HD5870 is 450W. e.g., https://www.ebay.com/itm/Dell-XPS-8700-8900-24-Pin-465W-Power-Supply-Unit-D460AM-02-HMCPC-Tested/264206754201?hash=item3d83f32199:g:GnYAAOSw35hcbfAt
This is considered a mini card and yes absolutely it will physically fit the MT case.
the caveat for RX550 is the card has a TDP of 50W and recommendation of 350 psu. It has no 6 pin power connector. Never mind the psu recommendation now. The question is whether optiplex PCIex16 slot can actually supply 50-75W. Look at motherboard does it have a 75W print next to the PCIex16 slot?
The optiplex spec of PCIex16 slot says it can support a 50W card according to the technical guide. 1GB AMD RADEONTM HD7570 is supported. It has max 50W. P.30/47 https://www.dell.com/downloads/global/products/optix/en/optiplex_7010_technical_guidebook.pdf
speedstep
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Posted May 9th, 2020 16:00
There is no such thing as "extra" aux pci-e power connectors. The stock power supply for ALL 7010 models is LESS THAN 300W which is why it has ZERO aux power connectors for video card.
Power supplies that work are 700W minimum. Thats why EVGA 700BR is recommended absolute BARE Minimum for the MINI TOWER model.
5870 might not physically fit into 7010 tower case because its too large to fit.
https://www.bestbuy.com/site/evga-700w-atx12v-eps12v-80-plus-bronze-power-supply-dc-dc-technology-black/6346160.p?skuId=6346160