I believe the answer is
"My parts are"..."hand in hand in a line"
refers to how the lines of the poem in between these two, are "parts" of the word, which go "hand-in-hand"
To expand on that, the hint means
that the parts of the word appear in the order they're presented ("they are affixed"), and that the letters from the end of each word will appear in the beginning of the next one ("the bounds of my neighboring parts are intermixed"). Admittedly, I'm not sure what "My ends they are betwixt" means, unless it's simply referring to how the lines referring to the parts of the word are "betwixt" the start and end lines of the poem.
As for each "part"...
"eternal rest" (death)
"pick up the tab"
"willing and able"
Put those together to get the answer.
"rest" + "(st)ar" + "(ar)t" + "(t)ab" + "(ab)le" = "restartable"
Which meshes well with the final clue:
Come again, you say?
Indeed, I may,
and then you will know my name, I pray.
restartable = you can start again, aka "I may come again"