1. If the enclosure for the ssd requires external power, then the y cable will be necessary, but if it's not one that requires an external power source, then you won't even need a Y cable. Just keep in mind people have had more issues using SSD's than traditional hard drives on the wii.
2. There are programs that allow you to adjust drive sleep, i know that Seagate has a specific one for their brand, but i'm not sure about others. It would still be receiving power while on, due to the external power source. So you'd have to unplug it from the external power source if you want the drive to turn off when you turn off your system.
3. All you gotta do is format the new drive in FAT32 with 32kb clusters, and copy/paste the contents from the old drive to the new drive. Since everything is in the proper folders already, this is the optimal solution to that question.
4. Like i said in point 1, the enclosure will only require a y cable if it's one that needs external power. The wii's usb ports put out enough energy to power an internally-powered drive off of a single cable, unlike the WiiU. So something like this here external SSD should work just fine without a y cable.

Edit: I just wanted to add that an ssd will make no difference in speed because the wii is limited to usb 2.0 speeds. So any speed difference will be negligible unless your current drive is experiencing issues that cause it to slow down