These pressing questions are meant to define what a personal commitment to God looks like in a life. The rest of the Psalm (consisting of four short verses) explains this. I see two main characteristics in the answer to these questions.
First, things are defined by personal character (15:1). In verse two we are shown three commitments in heart character: 1) walking blamelessly, 2) doing right, and 3) speaking truth from the heart. The point is not any kind of externalism. The righteous person does these things because of a deep personal commitment. The character flows from the deepest part of the heart.
Secondly, the person who can sojourn with God has visible social commitments demonstrated in righteousness. He does not slander. He does not do wrong to neighbors. He does not turn on friends and loved ones (Psalm 15:3). He despises evil but honors those who love God. When he makes a commitment, he will go the distance at cost to himself (Psalm 15:4). He does not seek personal gain in helping others. It is never about the money. He is not able to be bought (Psalm 15:5).
The person whose heart is changed by God will show that change in the lives of the people he knows. The world is a better place when God's people are in it. Neighborhoods are better. Communities connect. Relationships flourish. Cities thrive. Loving God brings civility and stability. This means that the gospel and the powerful heart change that it brings is essential to social stability and change.