Good question. Let me see..
For me, it has to do with feeling (as described above) and it has to do with common sense. By feeling, I'd have to experience it first hand and live that experience (do something, as in prayer, ritual), to be involved to know my feelings are concrete or are they fluff.
With God, being to me God is life, it is common sense. It is without "God" there is no life. Rather than God creating life, He (if you like) IS life. So my life, your life, everything living and that exists are God. (the personification of life not pantheism).
On that note you ask:
1. Does it mean that you are fully convinced?
Yes. I am fully convinced because I know that I am breathing, my heart is beating, and so forth and I and no human can create this out of thin air. I know because I can see creation just by watching a baby being born. I believe we come from the water and I can see God in the waters as we come from "Him" as He is the water that nourishes us and births us.
I know because I have experienced and feel the blessings and warnings that my deceased family has given me as I start talking with them more. The communication between us is supported by the spirit that keeps us together as a family--and to many people ,that spirit is God.
Do you know because of some specific evidence? (If so, please share)
Yes. In addition to above, when I see someone who lives in their faith, their behavior changes, their speech does, and how they see life. These are evidence that the spirit (or however one calls him or her or it) is working through that person to live.
I see evidence in the people in history who talk about their own experience with their God.
I see the evidence more strongly in how people change rather than written books.
Is it just strong faith?
It is strong faith.
Something else?
Knowing God exists is to have a relationship with life, nothing more.