@elmichel I'm going to respectfully disagree with @danj on his advice on checking the battery status. If the battery is swollen, it doesn't matter how well it's working (or not); it needs to be replaced. Period. A swollen battery indicates that gasses are building up inside due to the chemical reactions no longer working correctly. The condition will only get worse, and can eventually be catastrophic; you remember the Galaxy Note 7 debacle? Yeah, they can quite literally explode.
That's not all that likely to happen with a slightly swollen battery, but care does need to be taken in handling it.
So your first step is indeed to replace the battery, then evaluate whether that also fixes the trackpad.