I've tried a few breakout boards and my favourite is the WEMOS D1 Mini. All you need to supply is a MicroUSB cable. You can get them shipped from China for under $3: https://wiki.wemos.cc/products:d1:d1_mini
So now my kids have 2 wifi-tablet-controlled trucks with very custom programming (for N-point turns, playing music through the motor windings, headlight blinking, etc.) and the ability to update the program over the air at any time. It's basically magic. I spent only about $8 per truck.
Yeah, the WEMOS D1 is the real deal. I love it so much that I bought a pile of them before I've even decided what I want to use them for.
My second favorite ESP8266 hardware are the "Sonoff" brand WLAN mains power switches. I especially like the Sonoff S20 for its form factor, and the Sonoff POW for it's power metering capability. They contain an ESP8266 and can be reflashed with different firmware that does not need the vendors cloud services to work.
The actual smallest I've seen is one I made: https://github.com/skorokithakis/tiny-ESP8266-breakout
Big warning I'd make, though, is that the D1 has two revisions - the R1 (rarely described as such of course), and the R2.
You definitely want the R2. However, many ebay vendors are still selling R1's.
Yes and no. ESP8266 would typically survive 5V applied on any pin, and sometimes even sort-of work with 5V power applied (although with much higher power consumption and no WiFI).
But there is a catch. Pretty much all CMOS chips (ESP8266 inclided) have body diodes from each IO pin to power/ground. This means - you can't have voltage higher than VCC+0.6v or lower than GND-0.6v on any pin. So what happens when you have ESP8266 powered from 3.3v and connected to 5V I/O is - ESP will get (partially) powered via I/O pin. Depending on what 5V part is connected to that I/O, that might be bad for that part (excess of I/O sink current). And in extreme cases (5V I/O pin that can deliver high current, or ESP is in deep sleep), this will supply current into 3.3V VCC rail, bringing it up as high as 5 - 0.6 = 4.4v potentially. Which could be bad for anything else on that rail.
As a practical example - let's say we have ESP8266 powered by a switching supply with MP2307DN [1] chip delivering 3.3v, and connected to a 5v AVR. In normal operation, ESP takes ~70mA, which exceeds 40mA driving capability of AVR, so it's all good [2]. However once ESP8266 goes to deep sleep, all 40mA from AVR's I/O go via ESP's body diode back to MP2307DN switch, which usually would fail catastrophically and end up shorting input to output, supplying 12 (or whatever the input is) volts to ESP's VCC. Which is likely to fail as well, sending 12v further and frying the whole thing.
[1] MP2307DN is widely and cheaply available on "Mini360 DC-DC buck convertor" boards and it really doesn't like current being back-fed into its output
[2] Except I/O drive transistor in AVR is overloaded, which might or might not lead to its failure eventually.
I'm pretty sure that its the work done to make these into $3 wifi arduinos that is what made them popular. That's why wemos literally sells one that is shaped like, and has the same header spacing as, an arduino uno.