What's the requirement to give her sleeping pills/numbing skin? Do i need to max the heart? I can take her on a date yet she's still suspicious
None of those things influence her relationship level.
To raise your relationship, thus allow you to go on 'dates' with her (really just jogging together), you need to either
A) Buy her gifts and give them to her before sleeping
B) Pass skill checks during the day with her, such as helping her study, picking the lock on her door for her, or talking to her while she watches TV. Only one of these events can happen per day, so it's not a quick process.
Once you've reached a high enough relationship with her, you will automatically get an event where she joins you on a jog, regardless if you've been jogging or not. From that point on, as long as your relationship doesn't dip too low, you cna invite her on jogs where she might assist in helping you fight as well as randomly give you gifts when you meet her during the day.
There are a few more events tied to relationship, but the jogging is the only substantial one at this point in the game.