Your questions are not uncommon for a person with a new orchid. Trying to figure out exactly how to care for its needs is usually enough to deter most people from buying them in the first place. Orchids have been bred to be houseplants, so their care and maintenance is easier than you might imagine.
First, the flower stem dying back is a natural process. In the case of a Dendrobium, you should remove it after it is completely dead. When the stem is dead, it will look like a brown stick, and due to lack of nourishment from the plant, it will shrivel a bit. The stem will die from the tip back to the plant over the course of two to four weeks. Once the entire stem has turned brown, use a pair of sharp scissors to remove it. Generally, you should leave about 1/2" of the stem or enough so that it is not removed directly at the plant, and there is not obvious dead growth showing.
In general, orchids need to be planted in bark chips. In an attempt to cut down the cost of orchid crops, some growers will use other, less desirable media like foam peanuts, wool, or even clay pot pieces. While these do work temparaily, they are not meant for long-term orchid growth. Your orchid sounds much the same.
Since it has stopped blooming, now would be a good time to repot. Repotting is extremely stressful for any plant, and orchids are no exception. When you repot, take care not to damage the roots too much. They are fragile and will break easily. To help with this, set the plant in water (pot and all) for about an hour or two. This will allow the roots to take in water, and they will be more flexible.
Dendrobiums tend to be top-heavy plants. I would recommend one of the following:
1) A clay pot - This will add base weight, and some of these pots are fairly attractive for display. One concern with clay pots is that the roots tend to adhere to the clay more than plastic.
2) A plastic pot - The roots tend not to stick as tightly to plastic making it easier to repot later. However, some of these pots are relatively light. You might get some lava rock to mix in with your media or clay pot shards to put at the bottom of the pot to add weight.
3) A plastic pot inside a clay pot - Buy a nice clay pot and find a cheap plastic pot that will fit inside it. Plant the orchid in the plastic pot and set the plastic pot inside the clay pot. You may need to get a fastener to secure the two pots together.
When you repot, chose a pot about an inch larger than the current one. This is not always an easy task considering most pots are sold in even sizes (i.e. 2", 4", 6", etc.) Try to loosen the root ball a little with your fingers. When an orchid has been in a pot too long, the roots form a dense mat around the edge of the pot. Loosening the root ball enables new roots to extend beyond the current root ball after repotting. This will enable the roots to spread for plant stability and to encourage growth.
The bark chips come in a variety of sizes and textures. Most chips come as mixes of bark, moss, charcoal, lava rock, perlite, and other materials. I usually pick a fine to moderately coarse mix. These generally hold more water than the coarse mixes. A dendrobium is quicker to stabilize itself in a fine mix. I have actually had the plant fall over (in the pot) in a coarse mix just after repotting. I will sometimes add peat moss to the mix to aid in water retention. Keep in mind that a little bit goes a long way. Too much water in the mix and around the roots will cause root rot, and the mix will break down more quickly.
Dendrobiums are good starter orchids. They are easy to grow, and they bounce back after neglect. Dendrobiums like water and should not be allowed to dry out entirely. If you use peat moss, water once per week, otherwise two to three times per week is sufficient. An occasional soaking never hurts. With Dendrobiums, I have found that the normal humidity in my house (about 60%+) works fine. If the humidity in your house is less, you might consider misting on a weekly basis (daily if that proves insufficient). You can apply fertilizer biweekly during the growing season and monthly during the winter months.
With a little TLC, your orchid will grow, thrive and provide months of blooms at a time. The most important thing to remember is that you do not have to constantly worry about your orchid to make it grow. A little water and a bit of light (not too intense) will make it happy.