Why melting temperature of polymers depends on the prior crystallization temperature? This graph should show that the melting temperature of a polymer rises when the crystallization temperature of a polymer was higher (the melting temperature of a 100% crystalline polymer should be equal to crystallization temperature). Basically, the melting temperature depends on the history of the crystallization.

I'm not sure how I should explain this to myself. For now, I'd say that the higher crystallization temperature would mean slower crystallization and higher crystallinity, thus higher temperature required for melting.
 A: I think you have basically answered your own question. I will expand on it here:
The degree of crystallinity of a polymer will differ, depending on a number of factors. Polymers of higher crystallinity tend to have longer folding lengths.
In general, the longer the length of the individual 'crystals', the higher the melting point will be (so as you state in your question, higher crystallinity implies a higher melting temperature). The maximum melting temperature will occur at the maximum crystallisation, i.e in the idealised case where the polymer crystallises into a single crystal. If we denote this hypothetical temperature as $T_m^0$ then the relation is something like
$T_m = T_m^0\left( 1- \frac{2\sigma_e}{lh_f}\right)$,
where $T_m$ is the melting temperature, $\sigma_e$ is the surface free energy per fold, $h_f$ is the enthalpy of fusion, and $l$ is the length of the individual 'crystals', or ordered regions (also called lamellae, I think). We see that if the length $l$ increases the melting temperature increases also.
As to the question in the title: the melting temperature depends on the prior crystallisation temperature to the extent that the crystallisation temperature influenced the degree of crystallinity of the polymer.
Further reading
Melting Temperature and Change of Lamellar
Thickness with Time for Bulk Polyethylene
and 
Okui, N., 1990. Relationship between crystallization temperature and melting temperature in crystalline materials. Journal of Materials Science, 25(3), pp.1623-1631. (I'm afraid I couldn't find a link for this one).
