It's interesting that several of these consonant-ending feminine words are words that in Russian end with a feminine soft sign. I wonder if Russian borrowed them and added the soft sign because they were feminine, or if Czech took them and simply retained the femininity that was caused by the soft sign.
In Czech, both "c" [t͡s] and "č" [t͡ʃ] are voiceless affricates. The first one is alveolar, the second one postalveolar.
For orthography issues, they are both considered so called "soft consonants", that means that they are always followed by "i" and not "y" in writing - except for words of foreign origin (e.g. cyklista) and in case of "c" also some plural forms of masculine inanimate - e.g. tácy (trays), puncy (hallmarks), kecy (rubbish, nonsense) etc.
In Czech you have "č" and "c". The letter "č" is pronounced just like the "Cz" is in the word "Czech" in English (or like "ch" is in the word "chocolate"). The letter "c" is pronounced like "ts" in the English (well, japanese, but anyway) word "tsunami" or like the letter "z" is pronounced (usually) in German and Italian.
I don't know the linguistic terms, so I can't tell you which is affricate and which is fricative and I'm not sure what you mean by "soft". Czech has no soft signs (which I think look like lowercase b in the Russian alphabet) and I don't know how those are supposed to be pronounced (I don't speak any Russian). Maybe Russians write the Czech letter "č" like "c" (if written in our alphabet) plus the soft sign?
I guess it would be correct to say that in that specific case but it would also not be particularly clear what you want to say (if all you say are these two words). It would be grammatically correct, but it would be more natural and comprehensible to say something like:"Je to jinou noc" (i.e "it is on a different night").