I think it's necessary that the bitcoin rate be more stable
I'm not so sure about that. If merchants are using payment solutions to exchange instantly, then it's not really a big deal. Also, at that rate, then you've got a much higher velocity of money, which would put upwards pressure on the purchasing power of bitcoins. If anything, merchants would tend to hold what they didn't need for immediate liquidity and they'd welcome BTC purchases.
However, that's again predicated on enough merchants accepting it for it to be used much.
Yes, with due time this will occur. Even though Bitcoin is incredibly strong, growing and established for 8 years....you gotta remember that its still not at 1% adoption of all people yet.
Once it hits around 1.5%, and then a few years later when it hits 3% global adoption, then that pretty much equals mainstream.