Does repeating the salutation invalidate the prayer, such as saying "As-salamu alaykum" and then repeating "As-salamu alaykum wa rahmatullahi"?

It does not harm to limit the salutation (taslim (definition)) at the end of prayer to "Assalamu alaykum" without mentioning "wa rahmatullahi." The obligatory part of the salutation is "Assalamu alaykum," while "wa rahmatullah" is sunnah (definition) (recommended) and not obligatory. There are four forms of salutation authentically reported from the Prophet (peace be upon him), among them is limiting it to "Assalamu alaykum" to the right side only.