I too am delighted to find so many examples of JA's humour in her letters. She can be wry, ironic, and sometimes very caustic indeed. Her lack of "trite sentimentality" (as you put it) is one of the reasons why I love her work so much.

I find your second quote very telling. Here's the passage in full:-

"I have taken your hint, slight as it was, and have written to Mrs. Knight, and most sincerely do I hope it will not be in vain. I cannot endure the idea of her giving away her own wheel, and have told her no more than the truth, in saying that I could never use it with comfort. I had a great mind to add that, if she persisted in giving it, I would spin nothing with it but a rope to hang myself, but I was afraid of making it appear less serious matter of feeling than it really is."

I infer that Mrs Knight wishes to give her spinning wheel to JA, and that there is some considerable personal significance about that particular wheel which makes JA uncomfortable at the idea of taking it.

We see a flash of JA's propensity for biting humour in "spin nothing but a rope to hang myself". JA uses similar expressions elsewhere in her letters (eg Brabourne XIV and XXVII), and I read them as simply a bit of dramatic overstatement for the sake of a laugh, rather than having any really dark significance.

But JA is aware that her humour can wound and is sincerely tender for Mrs Knight's feelings, so she shares the thought only with her sister, as she recognises that Mr's Knight's offer involves matters of deep feeling.