I'm designing a system that has a camera (Ximea: MQ042RG-CM) that uses a female USB3.0 micro-B connector. I could connect to my computer through a male end, but I want to minimize the number of connectors I use in my system. I see that the USB 3.0 micro-B has 10 pins. Are some of them not crucial? I read that 5 of those pins are only there to make the port backwards compatible, but if I attempt to connect my computer to the camera using only the other 5, will it be enough? Does it depend on the camera? My computer supports USB 3.0.
No, you can't skip the original 4-pins, the ground pin is power ground, and you can't "skip" the VBUS.
Presence of VBUS indicates "cable/host" attach event, without "VBUS becomes present" (see Figure 9-1 in USB 3.2 specifications, page 317) the device won't proceed with Rx-detect and link training.
The only two wires you can possibly skip are D+ and D- wires, since USB 3.x connect/detect goes first, and USB 2.0 connect starts only if USB 3.x fails.
ADDITION: The pinout in OP is also oversimplified. In the original four-pins the ground pin is designated as "Ground for power return", while the additional ground in 5-pin added section is designated as "Ground for signal return", although they might be tied together. So you can't skip "Ground for power return", in some cases this might not work.