FJ: Buddhists seek refuge in the Buddha. Do people of other religions seek refuge?
What does it mean to “seek refuge”?  A refuge is a place where we can go for comfort, where we know we won’t be harmed.  Where we can rest and recuperate from the hardships of life.  It’s a place where we can be alone with ourselves. This “place” can be a physical place, like a church or temple, or bedroom, or meditation room, or the car we drive.  It can also be metaphorical – we can take refuge in a thought, a poem, or music.  When we take refuge in the Buddha we speak metaphorically.  Buddha is a metaphor for enlightenment – the awareness of our Essential Self. To take refuge in the Buddha we take refuge in our Essential Self – our true being. 
Other religions have similar metaphors for the same thing, and they also all have mystical traditions that involve meditation.  In Christianity we are familiar with the phrase “in the name of the Father, and of the Son, and of the Holy Spirit” referring to the three persons of the Holy Spirit (Buddhism has an exact parallel, the trikaya:  the Dharmakāya, the Sambhogakāya, the Nirmāṇakāya).  We implicitly take refuge in the three-in-the-one when we invoke the trikaya.  The Psalms refer to taking refuge often: Psalms 62:6-7: “He only is my rock and my salvation, My stronghold; I shall not be shaken. On God my salvation and my glory rest; The rock of my strength, my refuge is in God.”  And Psalm 46:1: “God is our refuge and strength, A very present help in trouble.” In Hindism, too, we have from the Kularvnava Tantra: “All fear of distress, grief, avarice, delusion and bewilderment exist only as long as one does not take refuge in the satguru.” In Islam, from the Qur’an, 41:36:  “And if there comes to you from Satan an evil suggestion, then seek refuge in Allah. Indeed, He is the Hearing, the Knowing.”
The closer we look at all religions, the more similar they all seem.
FJ: Is meditation characteristic only of Buddhism?
All religions have mystical traditions.  The principal tool used by all mystics is meditation. Some traditions are more "out in the open" with meditation, like Zen/Chan is with Chinese, Japanese, and Korean Buddhism.  While Christianity doesn’t emphasize meditation, there are many Christian mystics who practiced meditation--among the more famous are St. Catherine of Siena, Meister Eckhart, Hildegard Of Bingen, St. John Of The Cross, and Thomas Merton.  Islam too has many famous mystics, from Abdul-Qadir Gilani and Jalaaluddin Muhammad Rumi to Hazrat Shah Paran and Yusuf Hamdani.  Name any religion and you can name many famous mystics who practiced meditation.
FJ: How do we apply meditation in our daily life?
Meditation isn’t something we “apply” – it’s an activity we engage in.  Some people go the gym to exercise, or read a book to learn about something, or eat a meal to satisfy hunger.  Some people meditate.  Every activity we do serves some purpose--we do it to satisfy a yearning or desire for something.  We meditate to satisfy the desire for residing with Self, for being whole.  Once we learn to truly meditate, with continued meditation we gain the ability to carry with us that connection of wholeness in our daily life.
FJ: How do we know if we're making progress in meditation ?
Progress happens naturally when we attend to eradicating hindrances.  Signs of progress include more vivid, and archetypal dreams, and a greater awareness of one’s surroundings, feelings and thoughts.  Many people become agitated and irritated by other people, become moody, and may develop insomnia at the beginning.  While these may not sound like signs of progress you would expect, they are natural occurrences as the mind begins the process of emptying itself, releasing previously unconscious emotional contents into our consciousness.
There is nothing like true meditation and once it happens to us there is no mistaking it for anything else.  The challenge for many people is getting to that point that we are actually in a state of meditation.  There are many types of obstacles.  Most Buddhists are familiar with the “Five Hindrances”: sensory desire (kāmacchanda), ill-will (vyāpāda), sloth (thīna-middha), restlessness (uddhacca-kukkucca), and lack of conviction (vicikicchā).  But when we have adequate motivation, nothing will get in our way.
FJ: How do we quiet the mind?
There are many exercises to help with this and they all involve concentration to focus the mind. We might focus the mind on a thought, or a sensation such as sound or smell or our heartbeat, or by using a common zen training technique called the hua-tou. 
FJ: What is the effect of meditation?
This is beyond the scope of a short answer. There are many effects, and experiences, generated from meditation.  Emotional effects range from immense tranquility to ecstatic joy; physical effects range from perfect relaxation to bursting energy; mental effects range from clarity of thought to the falling away of all questions and uncertainties.
FJ: Stillness is something we will come to treasure over time. Why ?
When the mind is focused there is clarity and with clarity comes stillness, even in the presence of motion.  It is like being grounded: as a tree is grounded to the earth by its roots, we are grounded to our Selves, no longer pushed this way and that by emotions and the ever-present ego.   Out of stillness the Self becomes observed and we can dwell there.  It's treasured because it allows us to live in harmony with all that is.
FJ: How do you discover a part of yourself that is always at peace?
The Self is always at peace.  The Self that is not our ego, but our True Self.  It’s our lack of awareness of Self that moves us away from peace.  It is through meditaion practices that we discover that essential Self. 
FJ: When is the best time to meditate and how often and how long should one meditate?
Early morning and before bedtime are good times, as are any other times during tdhe day that is convenient and we are rested and not full of food (a full stomach disrupts the process).  The length of time need not be specified.  People who enter a state of meditation lose track of time and will easily sit for hours.  As you are learning to mediate, sit for as long as it's comfortable and you can maintain focused concentration without tiring.  Take breaks as needed. 
FJ: What is death?  What is birth or rather what is it that appears when there is birth, what is it that disappears when someone dies?
The mind creates elaborate concepts of birth and death through discriminative thought.  When there is no discriminative thought, there is no birth and no death, just being.  There is no longer fear of death because existence is seen as pure Being.  There is no birth because all that is, is.  The idea of past and future, birth and death, these are manifestations of a mind seeing separateness where, in reality, there is unity.  When the mind moves, and attaches to those motions, ideas and opinions and beliefs arise.  When the mind moves without those attachments, there is no fixed reality to its motion because it is always changing, always in flux.  Perceived reality, then, is no longer dependent.  The nature of reality is flux itself.  As Heraclitus famously said, "All is flux".  From the mystic's point of view, there is no such thing as birth or death.