To put what ibisc and mizinamo together, ym- is added to give a reflexive meaning to a variety of verbs:
golchi'r car wash the car
ymolchi wash (yourself)
You ask if it stays the same for different persons because mizinamo's examples do change
Je me lave I wash (myself)
Tu te laves You wash (yourself)
Elle/il/on/elles/ils se lave(nt) she/he/one/they wash her/him/oneself / themselves
The answer is that ym does not change as it is not a pronoun - or at least it isn't in Modern Welsh. It is just a marker that changes a verb from transitive to intransitive, although I have no idea what it was originally.
It is not ambiguous. English is a very flexible language where you can say all sorts of things you would not get away with in other languages, but one thing you cannot do is use a transitive verb without an object. You cannot say I made or I cut. So when you find a verb without an object you know it is intransitive/reflexive - ymolchi rather than golchi.
There are some apparent exceptions such as eat, paint, but these are things you would not normally do to yourself, so there is no confusion, and some dictionaries classify this as an intransitive use.