'Perhap' is the Middle English word (11th to 15th century English) from which the modern day 'perhaps' originated. It came from the latin 'per' meaning 'by' and 'hap' meaning 'chance'. We don't use it anymore but it can be found in Middle English literature.

I don't know about "perhap" but I remember seeing the word "mayhap" in Shakespeare or something old like that. However, I don't think that was short for "mayhaps," which was probably never a real word. Either way, "mayhap" is not a word anyone uses anymore. http://www.merriam-webster.com/dictionary/mayhap

Nope. Sounds plain silly.

No, never.