As long as the opamp is supplied from the 5v rail and ground there is no need to worry about it's output exceeding 5v. The lm358 isn't a rail to rail opamp, so it will never go all the way to 0 or 5v, it will be somewhere in between. (unless you run a dual supply like +12v / -12v, then you could get 0 and 5) The second opamp is just a voltage follower, whatever is on the input you get on the output, it buffers the first op-amp. It's cheap insurance against noise / interference from the adc pin and everything in between. Note: if we're working with different op-amps or voltage supplies the resistor values may be very different.
According to the k-type chart 1C should be 0.039MV not 39MV. Yikes! Too many zeros. I think it would be 0.000039V (I'm not arguing about 39 to 41, it's just what my chart said, it's minuscule) you're right on the 500C (my chart says 20.644mv or 0.020644V) I'm really only interested in soldering temperatures (100-500C ideal, 200-400C acceptable), that's where it needs to work well. If it works outside that great, but it's really not very important to me.so for 1 C the probe output (K type), and opamp input is 41mV 0,0041V, and for tops, lets indulge, 500 C, the output is 0,0205V.
My breadboard tests have shown (considerably) higher mv readings than the chart specifies, so I'm not sure what to make of it. You might try the non-inverting design (I think it's a lot like the Russian design) in the ltspice files linked a few posts back. It may work better at lower temperatures, perhaps even down around room temp. It may need some resistor tweaks. If you don't have ltspice it's free from linear technologies (google it). It runs on windows, but I think it works in linux under wine.
The reason for the extra temp sensor is to compensate for the "other" thermocouple(s). A thermocouple is just two different metals (wires) joined together, there is the joint we want to read and one (or more) at the other end of the wire, that will produce a voltage changing with temperature (generally room temperature). I'm not sure how much difference it makes, but that's what it's for.
I learn more when things don't work as expected than when they do, so have fun!