Is this possible on one motherboard? Can you have a mixture of non-RAID and RAID drives?
Yes, it's possible and done quite frequently.
Set your SATA ports to RAID mode, connect only your SSD, and install your O/S on it. AHCI mode is a subset of RAID mode so your SSD will automatically default to AHCI mode since it is not part of an actual RAID array.
After you've installed your O/S you can then shut down your system, connect your HDDs, boot into your motherboard's RAID BIOS, and create your RAID-0 array.