Question: In Russian, verbs in the past have gender information attached to them, so that “я спросил” implies that the asker was male, whereas “я спросила” comes from a female.

Why no other tenses have this trait, or why verbs were even chosen to differ by speaker's gender?

Answer: 
Because historically what we call past in modern Russian is perfect, and what we believe to be past forms of the verbs are in fact participles (adjectives formed from verbs).

Compare:

> 
> Он пел / она пела / оно пело (he / she / it has sung)
> 
> 
> Он бел / она бела / оно бело (he / she / it is white)
> 
> 
> 

In old Russian there was a number of other past tenses. The most used of them, the aorist, did not change by gender.
Good question. I guess for the same reason why in English you say "ten apples" and not "ten apple", you have said "ten" already, why add "s"? :) If I am not mistaken, in Chinese they say "ten apple". Historical, etymological reasons. Same in Spanish, even worse, the words are changed not only according to gender, but also according to tense and if its single or plural. English is so much easier :)