Solicit vs. request
You could interchange them, (i would advise against it though), they have slight nuance differences:
'Request' means you need or want to obtain something from someone
'Solicit' means that you will continually go out of your way to obtain something from someone [more persistent, coercion]
Nowadays, i believe that 'Solicit' has more of a negative connotation. It is usually used (in Australia) in negative situations/ crimes --> "He solicited sex" (not a 'request', implies coercion or influencing someone) "he kept trying to solicit me " (into doing something, again = a negative use). Request is more polite and positive.
"They solicited for a respond" --> something a lawyer or police officer may say something like this, once again it's more persistent, it may seem that the person may not actually WANT to respond. Although not a 'crime', it can imply someone influencing or coercing someone in order to get an answer.
"The teacher solicited me to open the classroom if he will be late for the first lesson" --> This can imply that the teacher coerced you into opening the classroom, maybe it was not something you wanted to do.
I hope this helps.