1) When you pray do you pray with belief and sincerity?

2) Or do you pray knowing that God is hearing your prayer, and in hopes that you will witness the hand of God at work? (not necessarily believing when you do it)

When I pray...I am usually not noticing God's presence around me or anything. And this is, to me at least, is what constitutes belief. Usually I pray and count on the fact that God is hearing my prayer. I don't necessarily sense belief when I am doing it. At the same time, I am not really doubting either. Does this make sense or no?

1) Yes
2) God, of course, being God, and therefore omnipotent and omniscient, knows our needs even before we express them. Though it is good to ask Him for our needs.

You seem to be referring here specifically to prayer of petition.

Jesus said Blessed are those who do not see and yet believe. We cannot see God but sometimes He gives us a feeling of the assurance of His Presence, though this is not at all essential to our Faith. Gazing on a beautiful image of Jesus and His Holy Mother can assist us.

Jesus also tells us to PRAY ALWAYS. That means that He will be on our minds and in our hearts in every waking moment. This is what must be working towards.