>Do Russians feel that certain rhymes are "cheap" or otherwise unworthy?
Yes. Rhyming verbs with verbs is considered so basic only novice poets do it. I'm pretty sure there are some examples of such rhymes being used in classic poetry, but they are always used in extreme moderation. This is because they are very easy to construct. So easy in fact, they are mostly not used in children's limericks.
Yes, but more creative people would be making something new, even if most would use it for nothing. No matter the amount of additional innovators, the total would still be bigger than it is now.