it is similar to how in Spanish "no" before a verb negates the action but alone means a general negation "no". I guess this is where Zamenhof got Esperanto's "ne" from. I believe I read a biography which said that his first language was Russian (even though he lived in Poland)... fun fact.
I'm native Russian, so may be I can help you ;)
Listen to this file: https://en.wikipedia.org/wiki/File:Ru-%D0%BF%D0%BB%D1%8B%D1%82%D1%8C.ogg (it's the word "плыть" = "to swim", it has only one vowel, and that vowel is Ы).
Or you can make grimaces:
Smile as wide as you can and, still smiling, try to say И (like wEEd, spEEd, swEEt). The sound should be more long and delicate/tender than English. That would be perfect Russian И.
Put down both corners of your lips while saying И (by muscles, not manually).
Next stuck out your jaw. I mean, your lower teeth should be more forward than upper ones. Well, have you seen orcs in The Elder Scrolls games? They have jaws like this =)
Change your voice to sound not too high. And, may be, add some apathetic / sad / argesive intonations.
After all that grimaces the sound И will be very similar to Ы.
Of course, Russians never make such grimaces to pronounce Ы, that's because they used to that sound. But the grimaces are very useful for the foreign students! For example, when we learn to pronounce "th"-sound, we literally stifle the tongue between the teeth (btw, that looks so stupid! XD), while native speakers only make a quick touch of teeth with no "I almost bite my tongue!' =)
P.S. Try to google something like "how to pronounce Russian Ы" and watch as much videos as you can, and always repeat every Russian word with Ы, even if your Ы is not perfect yet. The more you hear native Ы and try to repeat, the faster you'll be able to pronounce it correctly.
I was confused on this as well because I thought the easy way to remember the rule was the ending of the noun, ie. "мама" is a feminine word because of the "a" ending, which would also make папа a "feminine word" only because of it's ending. Therefore, I thought it was моя папа, which sounded correct to me... But I'm gathering that I'm totally wrong about this "rule".
It only could, if you put an emphasis on the words "не мой" in the sentence but then it would be like, if you met someone you are not sure if he is your father and you ask him, which is akward. "Не ты (ли) мой папа?" has the same meaning. Otherwise you have to say "Ты разве не мой папа?" или "Разве ты не мой отец?".