One reason bulbs can burn out quickly is if the voltage applied to them is higher than the expected voltage (120V in The USA). Wiring problems and bad transformers can cause the voltage to be out of spec. Another reason is if there is a loose connection somewhere, and the light flickers (causing unnecessary heating/cooling cycles). A third reason is if the light is in a confined space, and overheats.
In order to check the line voltage, you will need to use a voltmeter. The Kill-A-Watt is a very safe product to use to check the line voltage. Note that you should try a few different outlets, since there are generally two different phases of power (not really phases, but opposite polarities, 180 degrees apart) going into a typical house, and you need to check both of them.
Identifying a bad connection is trickier. Generally, you would see flickering lights, but to be sure you would want to use an oscilloscope or spectrum analyzer (expensive equipment). Sometimes a bad switch will cause a short flicker when the light is turned on. Replacing the light switches with new high quality switches could help.
Finally, have you noticed that the bulbs go out more quickly in certain places? Are those places more confined, causing the bulbs to heat up more? If so, you need to use lower wattage lights (such as CFL, halogen, or less bright incandescents) so that the bulb is less hot.
LED lights, halogens, and CFLs are all more efficient than incandescent lights. For the same amount of light output, they use less electricity, and generate less waste heat (keeping them cooler). In order to check if they meet your needs, compare the light output (in lumens) to that of the bulbs you are currently using. Don't go by the "equivalent" rating (such as 60W equivalent).... The amount of heat generated is nearly proportional to the wattage of the bulb (generally between 10 and 25 W). Another thing to look at is the color temperature, as measured in Kelvins. More orange lights will be around 2700 K, while more blue (daylight) bulbs will be around 5000 K.
PS: Advertised bulb lifetimes are usually assuming use of perhaps four hours per day. So, a seven year bulb may only last a bit over a year if it is left turned on continuously.
Almost all recessed light trim rings are held in place by spring clips that grip the interior of the recessed can. However, some of those clips have sharp prongs to better grip the can.
You often have to use some force and wiggle the trim ring downward to overcome the grip. Sometimes, if you get a little gap, you can use a flat bladed tool, small putty knife or flat screwdriver, to locate the clip and push it towards the center as you ease the ring downward. Sometimes rotating the trim as you pull downwards helps, even though the trim is not really screwed in.
The trims routinely scrape a groove in the cans as they come out, but this does not damage the can and will be covered when the trim is replaced.
Work carefully, because some cans are not well attached, especially old work cans that may not be anchored to framing. I would strongly recommend turning off the breaker to the can before working. Turning the switch off is not enough, since there may be live hot in the can, even if the wall switch is off.
Finally, I would check, before removing the trim, to make sure that the lens does not come off by itself. If the lens were removable, it probably would require a counterclockwise twist while pushing gently upward.
Best Answer
This looks like fixtures I have had, which are held in place by three screws. It's hard to be certain from the photo, but if it is like this, then you should loosen the screws by hand bit by bit until you are able to remove the globe. That should expose the bulb and the socket it's screwed into.
If not, then please climb up higher to take a photograph that shows the part of the globe that comes closest to the ceiling.