...will then travel by train to Budapest, Vienna and Prague.
Might be a good idea to travel Salzburg-Vienna-Budapest-Prague, thus avoid going leg Vienna-Budapest twice (all the trains from Salzburg to Budapest travel through Vienna), plus it will be a cheaper solution, since Salzburg-Vienna tickets start from 19 euro (discounted from OeBB bought in advance for specific train) or 25 euro (full fare, WestBahn), then Vienna-Budapest 19 euro (discounted bought in advance) or 36 euro (full fare) and finally Budapest-Prague (discounted from 19 euro)
...booking Budapest to Vienna is a challenge on the MAV website
Not sure if you've seen it, but we have a special step by step guide for buying tickets from MAV website - http://europetrainsguide.com/Advice/How-to-Buy-Train-Tickets/Hungary-MAV.html
, it can simplify the task a bit
not sure if the costs will be the same as Option 1.
No, discounted tickets are rather essentially cheaper, so in case you arrive at the station and try to purchase your tickets before departure then you'll pay more. In case of your original itinerary (Salzburg-Budapest-Vienna-Prague) it will be 79,6 euro (full fare Salzburg-Budapest), 27 euro (full fare Budapest-Vienna bought in Hungary) and 64,2 euro (Wien Meidling-Prague). Discounted tickets (--> bought in advance and for the specific train) instead add up to 39 euro (Salzburg-Budapest), 13 euro (Budapest-Vienna) and 29 euro (Vienna-Prague). Although all of the above discounted fares are the lowest available, while since your travel date is rather soon the cheapest ones might be already sold out for certain departures