first of all: The 3P now sounds ground! Very good work.

But I have to admit I have troubles with saving tones and sequences.

If I try to save a sequence everything looks good, but if I change to
another sequencer, the store this sequence, and try to load the first
sequence, then the first sequence is empty.

The procedure is:
*Tape
*Seq write (enter sequence edit mode)
*Tone1 (select sequence No.1)

No, this is not correct. Please study the menu map I sweated blood over making. Tape:Seq Write:Tone # will erase a sequence. This is the only way to clear a seq. To create or edit an existing seq press Seq Write:tone #.
The 3P seq write is always in edit mode when writing a seq. When you press Seq Write:Tone # the existing seq is not cleared. I have detailed this in the latest user manual.

If I try to save an edited tone, then the JX3P either does not store the
sound, and somethimes it jumps into some completely uncontrollable state
(some LEDs are lit or blink, no reaction on any button or key at all -
then only a hard power reset revives the synth).

This sounds like a mixture of versions, I had the same problem during testing. Ignore the files I emailed to you and download the v2 update from the web and reapply just the Boot v100b & Prog v200b files.