Hello Rosiegabrielle and welcome,
I cannot help with the Indian visa question, the answer seems to vary from one nationality to another and I am a Brit.
Yes an Indian registered bike can go into Nepal, I have seen it posted that there is a small daily fee and a limited number of days but also that you can go straight in with one without any fee or restriction, Nepali customs are very relaxed so it might vary from one entry point to another. It is some time since I entered from India but the eastern crossing near Darjeeling was good, the lady in charge on the Indian side gave me chai and the Nepalese chap gave me cucumber dipped in chilli all of which made for a pleasant crossing.
A Nepali visa is availalbe on entry.
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