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

* the Russian past tense is the past perfect
* the past perfect is the past tense form of the verb (eg *пел*) + the present tense form of the auxiliary verb
* the past tense form of the verb is a predicate adjective
* predicate adjectives requires gender/number agreement
* the auxiliary verb is the copula (*быть*)
* **the copula is dropped in modern Russian in the present tense**

(That dropped copula is what causes the confusion.)

So if we include the copula, as in older\* Russian:

> 
> Он есть пел.  
> 
>  Она есть пела.  
> 
>  Оно есть пело.  
> 
>  Они суть пели.  
> 
>  Мы есмы пели. 
> 
> 
> 

It is analogous to Ukrainian, Serbian and so on (and even Italian and French for many verbs).

It was true for Russian copula in any situation.

> 
> Он есть инженер. Погода есть хорошая.
> 
> 
> 

But in modern Russian:

> 
> Он ∅ пел.  
> 
>  Она ∅ пела.  
> 
>  Оно ∅ пело.  
> 
>  Они ∅ пели.  
> 
>  Мы ∅ пели.  
> 
>  Он ∅ инженер.  
> 
>  Погода ∅ хорошая.
> 
> 
> 

The copula is dropped, but the agreement is still required.

*How old is old?* [Wikipedia article on zero copula](https://en.wikipedia.org/wiki/Zero_copula#Russian): 

> 
> The present tense of the copula in Russian was in common use well into the 19th Century (as attested in the works of Fyodor Dostoyevsky) but is now used only for archaic effect.
> 
> 
>
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 :)