Can only fully matured sperm fertilize an egg?
As far as I know, sperm cells undergo maturation in the epididymis, where they become motile. As Eliane pointed out, this is a continuous process, so the male does not have to "wait", because there is continuously a batch of sperm cells that will shortly become mature. Furthermore, maternal factors are required for the process of capacitation in which the sperm cell becomes capable of fertilization and undergoes motility increase. 


An immature sperm cell is simply non-motile and even if present in the ejaculate, it will not be able to travel towards the fallopian tube to fertilize the oocyte. A sperm cell which did not go capacitation will also have no ability to fertilize an egg (that's why prior to IVF it's important to capacitate sperm). 

My sources:


https://kjcscientific.wordpress.com/project-excalibur/
Gilbert's Developmental Biology (2009)