You do need a SATA cable and yes run the power cable over. Removing it from the retainer is fine just make sure not to route it too close to a fan. After the HDD is installed make sure to open Disk Management to allocate and format the drive. This video will help.

The sata power and data cable to 5050 laptop style thin odd may be mini sata combo cable and thus incompatible with regular 2.5 or 3.5 sata drive.

you may need another Dell proprietary mini 8 pin to two sata cable for a second storage drive.

This is an example of the cable you could use.

Dell proprietary mini 8 or 6 pin sata pwr socket on motherboard is a mess in earlier models.  Some model uses 8 pin and some 6 pin, and some model use different pin for MT and SFF of same model (the later more modern Dell have standardized to 6 pin).  What is known is that 5050 MT and 5040 MT motherboard both use mini 8 pin sata power socket.  therefore the cable below applies to 5050 MT too.

https://www.comeap.com/COMeap-Motherboard-ATX-Mini-8-Pin-to-2X-Right-Angle-15-Pin-SATA-Hard-Drive-HDD-Power-Adapter-Cable-for-Dell-OptiPlex-Series-3020-5040-7020-9020-14-5-inch-37cm-p373043.html

Re: Coming out of that connector is a red wire and a black wire that go to the ODD.

you may be referring to the slim line sata for odd which can not be used for regular sata storage.

If the above is confusing to some reader, it helps to know that 5050 psu is Dell proprietary 12V only. psu only supplies white 6 pin and white 4 pin to mobo, both 12V only.  motherboard circuit then generates 3.3 and 5V which are needed for standard sata 2.5 ssd or 2.5/3.5 hdd.  these power come out of the mini 8 pin black socket (sata pwr) on mobo.

the red/black two wire slim sata power for odd is 12V/gnd.  the slim odd does not require 3.3 or 5V.

the other thing you may or may not notice is mobo has 4 sata data ports of different color, blue, two black, white.

use the blue one (sata_0) for boot drive or 2.5 ssd because by convention it is the fastest (sata 3.0 or 600 MB/s).  use the black one for data back up drive or hdd (these may be sata 2.0 or 300 MB/s).  white one (sata_1) may be the slowest (sata 1.0 or 150MB/s) and typically assigned to odd.