I know that some operations are defined for surreal numbers like addition or multiplication.
Are all operations for all "number systems" defined for surreal numbers in a way that they give same result as respective operations defined for those "number systems"?
No, surreal numbers do not contain all other number systems. They contain real numbers, which are a completion of rational numbers using the ordinary absolute value as a metric, but not $p$-adic numbers, which are a completion of rational numbers using the $p$-adic metric, where numbers are smaller if they are divisible by a higher power of $p$.