Yes, it is possible.
Yes, it is possible.
The SATA power from the hard drive is not designed for a GPU connection. You can't safely plug in a GPU power adapter into it.
The SATA port you highlighted isn't just passive—it actually supplies power to the SSD. One SATA connector is limited to about 40 watts, though they claim up to 55 watts officially. A 6-pin PCIe connector must deliver up to 75 watts for a video card, while an 8-pin needs 150 watts. To meet these requirements, you'd need an adapter with two SATA ports on one side and a 6-pin PCIe connector on the other.
The Molex connector you mentioned fits two SATA drives. You need something like this: https://www.amazon.com/Express-Molex-Pow...215&sr=8-2 I prefer a PSU with the right connectors. If it lacks them, it probably isn't powerful enough from the start.
It seems you're facing some challenges with your setup. Without two free SATA power cables, installing a PSU might be tricky. For a GPU choice, consider what you need—budget, compatibility, and performance. Look into mid-range options that fit your system requirements.