LED strip heating troubles
The &quot;wires&quot; on the strip are just thin flat printed circuit board strips, so they lack the plumpness that you need.  You can make all the LEDs heat up equally by hooking up the positive at one end and the negative at the other end of the strip, or you can reduce the currents on the super thin built in wires by tapping power in at several places over the length of the strip.  The latter is generally the better option although if you wanted to be a snob you could split the strip into multiple sections and do both.
As far as the voltage rating of your LEDs, you have voltage rated LEDS, which waste a bit of power by using a resistor to regulate current per series string.  12.2V isn't far enough over 12V that you'd be putting the LEDs at risk(especially if you add wiring to feed them properly as that lets them age the same), but underpowering them will actually increase their operational life as long as brightness is still adequate.  As you've been finding with your diodes, dropping the voltage by wasting the extra power is a non ideal plan.  The power your diodes waste is equal to the voltage drop over them multiplied by the current, and with 0.8V over each diode and a significant current, the diode is heated considerably.  You need to either cool the diode or divide the wasted power between multiple diodes so each doesn't heat up too much.  A better way would be to decrease output voltage, perhaps use a dimmer driver.  Your diode can survive the thermal abuse for some period of time, but it's lifetime will be shortened the hotter it gets.