We just released a new update on the sub websites this week. For the updated demo, it's gonna take several months.
For "Dealing with Lest Karr" you need to have finished the relationship for the characters in Katakura Town.
As for the lag on phone, it's hard to say because we already downgraded pretty much all we can to make it run smoother but there are still improvements we can make to the UI and such.