Hmm. I used to use a fountain pen sometimes. I haven’t in years. I should try again and see what i did. (Yes, I’m left-handed.) I think i tried to use absorbent password, though.
Your password is too weak. Please choose one that is less absorbent.
Mountain of Ink has a collection of fast drying inks. That isn’t perfect however faster drying ink will limit the chance of smearing.
Paper. 
As long as your absorbent paper password isn’t a blotter; those get everything backwards. Which will never work to let you in.
Can you hold the pen so the contact point of your hand with the paper is positioned under the line, if you follow me? I assume you have tried this and it is not comfortable? The pen itself does not have any chirality or handedness.
They would if their palms were resting on the same line they were writing. Cf. tutorials in Arabic calligraphy: the first lesson is on how to hold the pen and write the basic strikes.
You can use quick-drying ink, of course.
Either way, it is not necessary to jam the nib into the paper; no pressure should be required. (Perhaps it is more a matter of upstrokes vs. downstrokes as RTL versus LTR?)
Aw, you’re kind! Yep, I’m a lefty, and also a dork.
Sadly, I don’t have much help here. My handwriting was so bad as a kid that a specialist came in to help me. Unfortunately, my stubbornness was so intense that it didn’t help much. The only thing that really helped was learning to touch-type. To this day, my hand cramps up something fierce when I try to write without smearing any sort of ink-pen.
As for teaching handwriting to kids–this has never been my forte. I can teach ‘em to memorize poetry, to use spreadsheets, to devise plausible ecosystems, to discuss literary tropes–but handwriting? Oh boy, I hope they have another teacher who can do a better job with this.
This is true. When i played with a lot of pens, i used a very light hand, and the fountain pen nib doesn’t get pulled or pushed, it glides gently over the paper. That’s one thing i liked about using fountain pens.