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Buddha site visit itinerary help

I'm planning to travel following itinerary for 6 days, the intention of the travel is to visit sites related to Buddha. After searching in the internet I made following itinerary. Can someone help me with planning the travel.

Bodh Gaya -> Patna -> Kushinagar -> Lumbini -> Shrawasti -> Saranath

I'm having limited time and start and off points are fixed. However can make some changes in between. I'm having following points to clarify.

1. what would be the cheapest and safest way to travel in between above sites

2. Since some of the sites are located in Nepal, would it required to have Nepali visa also ?

3. where can I found best hotels nearby these sites.