There are a number of variables you need to consider in order to space them correctly.
First, how much light do you want? Brighter rooms require more lights. Related to this is the max wattage of your cans. You will need more 35W lights to achieve the same brightness you'd get with 50W cans.
What type of spread (angle of light ) do the bulbs you plan on using produce? Bulbs with a wider spread will cover a larger area but light will be more diffused.
The typical rule-of-thumb is 24" from each wall, and then 3-5 feet between cans. So in a 12' room you'd probably want two rows of lights. Considering it is a bedroom and you don't need it as bright as a kitchen, you can probably get away with 2 rows of 4 lights for a total of 8. If you are using 50W bulbs then you will have 400W of light in your room.
The height of your ceiling also plays a big role. Higher ceilings typically require more cans since the light diffuses more before it hits the floor.
Also take into consideration the position of ceiling joists; this might limit where and how many you can place.
I always recommend drawing out your design on graphing paper before you make your first cut. Find out the spread of your light (taking into account the ceiling height), and draw a circle under each can with a protractor. You want the light produced on the floor for each can to slightly overlap another light so that thre are no dead spots.
Make sure you check the ceiling joists before you start cutting!
I should also mention that if you are installing into an insulated area, make sure you use IC (insulation contact) rated cans!
Chances are that this fixture was installed at the time of the home's construction. It'll likely have two crossbars that were installed across the gap between the two adjacent roof trusses or floor joists:
Removing it would require access above the ceiling, or you can attempt to cut and disassemble the unit through the hole in the ceiling. If you can release any tabs that secure the can itself to the base plate you may be able to push the can up out of the way, allowing access to the crossbars.
Be aware that there's likely blown or batt insulation above and around the can. Take precautions to not get it in your eyes or any other sensitive or delicate places.
Best Answer
This question may be too subjective for this site, but to give you my input, I don't quite understand why people say to keep recessed lighting so far from the wall. It's supposed to be more of an accent-type light, so you really want to wash the walls, not the floor, with light, hence, move them closer to the wells. Then consider putting different types if fixtures elsewhere in the room, for example a pendant over a table or sitting area.