How do you replace a fusible link?

Can I just cut the rubber piece off and splice them all three together or does the rubber piece have to be where they come together?
Assuming the fusible wire is in good shape you should be able to cut the rubber off and splice the wires together. I would be sure to use a high amp splice preferable same type as OE. If when you remove the black rubber all the strands of wire are still attached to each other as the OE made it you could create a mold and over mold the OE splice area with appropriate black material. I think Eastwood sells such.

Keep in mind that if you shorten the fusible link wire you are changing its characteristics. My best guess 1/4 to 3/8 of in inch probably negligible.