Any country has different rules-also about how to take a dog with you-from near complete non-allowed till ´no problem´ In Scandinavia there are often some cars barred for any animal. In other country the beast need to be muzzled, have all injections/pass and sometimes must be in cage. There may be MAX size etc. You cannot reasonably expect normal travellers to know that all-most countries have clubs for doglovers who are usualy the best source onhowtodoit.

There is no Interrail pass for dogs.

You need to check with the train companies that you will use, what rules they have for dogs.