understanding WIFI range

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Some routers, some labeled travel routers, can be used as clients. You are right, boosting power may be of some help, but improved antennas are a better solution, effectively boosting power and receive sensitivity.

Directional antennas are best, and need not be overly directional, covering a sector for desired coverage, at some loss of coverage in directions where it isn't needed anyway.

I added a free reflector to the router. It provides improved coverage over a fairly broad area in the direction that I want. I have an L shaped house, with the router at the top of the L. Coverage with the EZ-12 is good throughout the house, even out to the bottom tip of the L, and on the deck inside the angle of the L.

EZ-12, printed on photo paper for thick stock, with aluminum foil glued to the sail, provides a substantial boost in signal. The netstumbler trace shows solid signal as I walked back to the router, a dropout as I blocked the router, taking off the reflector, and then the lower signal without the reflector, reduced, and fluctuating in level.

Make the tabs longer than the template drawing, for easier assembly.

For the most part, you are correct. If you change nothing but the output power of the AP, the connecting stations will still be limited by their output power. As a ham operator, I've found that the antenna is you best bet to improve performance in any king of tranceiver, be it a 2m radio or a wireless AP. Take that AP I mentioned above. If you add a directional (like the one you can make at freeantennas.com) antenna; This increases power by sending the majority of the signal in a 120-170 degree path (instead of 360 degrees) while also improving reception (gain) in that same arc. On my AP at the house, my laptop signal strength was at 60% and increased to

80% when I added the reflector (directional) element. They aren't scientific measurements, but performance improved.

You are correct though that if you put out 10watts thru the stock antenna (this would exceed the FCC legal limits by the way), the connecting station might hear you, but probably won't connect depending on its location. If you run a card with a true antenna on your connecting station, with correct placement it will also improve your connection to the AP. So if you point both antennas at each other, your range performance will increase even further. The better balanced your antenna is to the transmitter (SWR), the better your system will be. More power will make it to and thru the antenna.

Did I confuse you enough? I tend to ramble on from time to time.

By gosh, you have it correct. (To a point.) They not only increase the effective radiated power of the access point, but they also increase it's sensitivity. Making it more sensitive, gives the effect of making your remote unit more powerful as the access point is more able to see it.

If you increase the radiated power by using a higher gain antenna, then THAT way it will increase the received signal level.