What is the best possible SSD upgrade for an A1181 MacBook?

You may want to consider some other options here.

To answer your Questions, Yes, you can put in a SSD to replace your current SATA I (1.5 Gb/s) 2.5" 9.5 mm HD. And, you don't need to remove your optical drive. Or, make any modifications to the systems hardware.

But, you will have issues with a SSD in this system. This system is limited to Mac OS X 10.6 Snow Leopard and can't run anything newer. Why this is important is Snow Leopard does not offer TRIM support.

I would strongly recommend going with a Hybrid drive (SSHD) instead {i.e. Seagate}, here you gain the benefit of the SSD but not handicapped with the lack of TRIM as the drives firmware handles all of that and you get the performance you want.