Can we make unbalanced IQ mixer? E.g. using only two Schottky diodes.

I have seen some IQ mixers with malfunction of one or maybe even two Schottky diodes. So only 2 of 4 diodes actually worked fine, while others burned out/defective/etc. Although those mixers still gived some acceptable phase information, sometimes with noise or big difference in amplitude of I and Q channels. It gived me idea that on practice we can construct IQ mixer with only use of two Schottky diodes with 90 deg delay line. At some price...

sure. If you can make a mixer out of one diode, you can certainly make two mixers/phase shifter/combiners to make up an I/Q mixer. The topology for LO injection is more complicated, which is why people prefer balance pair or quad of diodes for each mixer. Also you have to deal with NO LO suppression at either the RF or IF ports

Basically you would set up the single diode in a way that it would sample the RF signal at the LO rate, and send that output to an IF port with a small storage capacitor there.

If a double-balanced mixer using four diodes lose two diodes, became a single-balanced mixer.
The advantages of a double-balanced mixer (with four diodes) over a single balanced mixer (with two diodes) are: better linearity, improved suppression of spurious products (all even-order products of the LO and/or the RF are suppressed), and better isolation between all ports.
The disadvantage is, the quad mixers requires a higher level LO drive, and needs two transformer baluns, when a two-diode mixer needs only one balun.