Power seems to be an important aspect in your design, so you might consider using different LEDs. Low-current LEDs for instance take 2mA instead of 20mA.
Or maybe 5mA through your existing LEDs will produce enough light for your taste.
Another angle would be to design your LED drive circuit for as low a voltage as you can. If you can design it for 3V and you use a switched power supply to create this 3V from 5V (or maybe 12V), you have saved 40% energy. This is especially effective when you use a battry, which produces a lower voltage over time.
I guess that when you want to iluminate (nearly) all LEDs you can get by with illuminating them each a bit less. This logic could be incorporated in your software, so the worst case current would be reduced.
You must design your circuitry for the worst case, but for battery life it might be more realistic to calculate with averaged cases. So get some more info on what you want to show when you need to know how a given battery will last.
As often, your one good question results in an avelange of questions, and some questioning of your basic assumptions. That's system design :)
Most of these strips are laid out in a repeating pattern of three LEDs and their current-limiting resistors along a flexible circuit. There will be scissors marks every 10 cm or so.
simulate this circuit – Schematic created using CircuitLab
Figure 1. Cut along the dotted line.
If it's this type of strip then the LEDs are effectively wired in series parallel along the length of the strip. You can connect the strips end to end but it should be clear from the schematic that you're actually connecting the individual strings of LEDs in parallel.
If you daisy-chain the strips then be aware that the first strip has to carry the current for all the downstream LEDs and the copper may overheat. You've no data sheet so you need to get a feel for what's normal temperature with one strip and add another, etc., until you feel it's getting too hot.
Alternatively wire each strip individually back to the PSU.
If you have a multimeter switch it to 10A DC and plug the red lead into the 10 A socket. Measure the current drawn by one string and work out how many strings your PSU can handle.
Update
The photo arrived after this post. Basic principle still stands but I can see a resistor per LED and the LEDs have six pins. You'll have to figure it out.
Calculating current requirement
simulate this circuit
Figure 2. Measuring LED strip current.
Hook up a multimeter as shown in Figure 2. and measure the current drawn by one strip. If using a digital multimeter then be sure to plug the red lead into the 10 A socket, and switch to 10 A range before powering up. The total amps will be the sum of all the individual amps readings.
Converting to watts
Let's say you measure 0.9 A. Power for that strip will be given by \$P = V \cdot I = 12 \cdot 0.9 = 10.8 W\$. You can then figure out how many of these strips you can power from a PSU of known wattage.
Your flashing LEDs indicate the power supply has shutdown - probably its thermal protection kicking in.
Plug your red lead back into the V socket before you finish. If you forget and hook it up to a battery, etc., a high current will flow, blow the internal fuse and possibly damage the internal shunt resistor.
Best Answer
Your calculations are correct. 354 LEDs at 20mA each is 7.08 A (354 * 0.02).
The thing to be careful about with LED strips is the voltage drop. The copper thickness on strips like the 5 meter one you have tends to be 0.5 to 1.0 oz, and the voltage drop is quite noticeable. In other words, if you supply voltage to one end of the strip (12V), the opposite end will only receive 10-11V. The instructions may indicate that you should supply power at both ends of the strip, which is a good idea. Personally, in my projects, I do not run strips longer than 2 meters with that type of copper.
Edit:
Looking at the datasheet for the LED strip, it is a 5 meter strip with 60 LEDs per meter. Each "LED" is actually a 5060 package SMD RGB assembly, which means there are three LEDs per package. The datasheet specifies the maximum If at 30mA (not 20) for each color, R, G, and B. (The characteristics are derived using a test If of 20mA, but the absolute maximum value is a worst-case scenario, which you should use in calculating power supply size.)
Unless I am reading it incorrectly, this means to light the strip at 100% brightness at white (all three colors), it will need 90mA per LED. That works out to 5.4A per meter, or 27A total.
Comments on the product page suggest that the strip only requires 6A. I suggest measuring to be sure. (Strips that I have require about 20mA per color, which is what made me think to come back and add this addendum.)