Yes, it does have a religious connotation but it also does simply mean wrongdoing and I think when you look at a lot of what is happening in the world, it seems an appropriate term. Or you could say evil instead. Sexual exploitation, misogyny, murder and genocide etc.
But as a Christian it's about acknowledging my own faults and how they fit into the bigger picture, seeking forgiveness, trying to do good and equally important, forgiving others. I believe God wants to reconcile us with Himself and with each other, so that means accepting and offering forgiveness and also restitution where appropriate.
Misusing religion and seeking power in order to oppress others whether it be under the name of Islam, Christianity or Communism or whatever, is also sinful/wrong. There's no perfect, utopian system that will erase the human tendency to be destructive though unfortunately.
I agree there are greater and lesser sins, but I view it as a continuum in a way. Even minor things can have significant ripples and ill-effects.
I don't think you need religion to be moral, but Christian thought has formed the basis of a lot of the ethics we now take for granted.
As for thinking in terms of the hereafter, yes that is wrong if that's how you treat your faith. Christians are called to be part of and build the Kingdom of God on earth and given lots of guidance on what that looks like. But if you believe you have a soul, it's important to have peace with your Creator and think about what path you may be on. It's also a great hope for Christians, and it's built on Jesus's own resurrection and promises to us. So the idea is to live a good life, seek forgiveness for our failings and look forward to 'going home' at the end of our earthly lives.