Router antennas (most of them) are most definitely omni-directional. It's a simple half-dipole as you see. Indeed tip #4 works specifically because you are taking some of the power radiated from one side of the antenna and reflecting it back, effectively creating a crude directional antenna (reception will be really poor on the other side of that beer can and better on the antenna side).
Anyone who has studied radio theory knows that there is an (amazingly complicated) formula for power output-to-power received. One of the main variables in the equation is the height of the antenna (either antenna). The voice intercept system we used in Saudi/Iraq/Kuwait in the Gulf War was installed on a helicopter. Nothing like putting your receiving antenna several thousand feet in the air to improve reception! Put the router in the highest place in the house, but do not invert (won't hurt, just won't do anything) the antenna and definitely don't lay it flat (your WiFi devices are looking for a vertically polarized signal, laying it flat will change it to a horizontally polarized signal). It is possible, however, that there will be a dead spot directly underneath the router if it's on the floor above you.
How about suggestions for covering multiple stories. Should the router be at the ceiling of the lower floor or upper floor. In each case should the antennae still be up or should they be inverted if placed on the ceiling of the upper floor/attic?
Also, is there a good, responsive signal strength meter which can be used to gauge signal. Most laptops have about 4 levels of signal, ipads have 3 or 4, and both are very, very laggy - to the point of being useless for real time mapping/verification. Ideally one which is affordable (or maybe a mobile app or laptop app with a real-time signal strength meter)?
You can get a better signal meter by using one of the tools mentioned in tip 8. I use inSSIDer (http://www.metageek.net/products/inssider/) which has a Mac and Windows version, but the two they listed are also good. You not only see the strength of your signal, but you see the strength of all of the other networks around. In my case, I saw that none of my neighbors are using the 5GHz band, so when I replaced my router, I got one that works up there and I've got clear signal while they all fight in 2.4GHz land.
Really, in a home with multiple stories, you should have a router on each floor. Router antennas (most of them) are not omni-directional, they send out signals forward from the router antenna and not backwards nor up and down.
Now, for connecting the routers together, I would advise running a wired connection between the routers to get the best speeds possible.
Actually most home-wifi routers that use "rubber duck" type external type antennas ARE ommi-directional antennas with a typical gain between 2-7 dBi. The RF pattern from these in an optimal unobstructed 3d space is similar to droopy doughnut.
Actually its not advised to connect router to router, unless it has a built-in AP mode.
Its best to just get a Repeater, or Wireless Access Point (AP) Unit.. If you want to go cheap, find an old router that is supported by DD-WRT, Tomato, or OpenWRT and set it as a AP/Repeater.
That's dumb. Get ONE router. Then get ONE switch. Then get as many wireless access points as necessary. Plug in accordingly. Having 3 routers because you have three floors is ridiculous. Why have three routing tables actively running, in a chain, in your internal network!? Can you imagine forwarding a port! Jeez!!