Have a look at the table in https://en.wikipedia.org/wiki/Colloquial_Welsh_morphology#Initial_consonant_mutation .
So the lenition/soft mutation is similar to eclipsis in Irish for unvoiced stops (p : b, t : d, c : g) but not for voiced stops, which become fricatives in Welsh (b : f /v/, d : dd, g : [nothing]) but nasal stops in Irish (b : m, d : n, g : ŋ); also, the sounds "m, rh, ll" are affected in Welsh but not in Irish.
"m : f" of the soft mutation is similar to what happens with lenition in Irish, as is "b : f, d : dd, g : [nothing]".
The behaviour of unvoiced stops under lenition in Irish is similar to what happens to them under aspirate mutation in Welsh (p : ph, c : ch), but "t" goes to "th" rather than to "h".
And the behaviour of voiced stops under eclipses in Irish is similar to the nasal mutation in Welsh: "b : m, d : n, g: ng", but Welsh also has nasal mutation for unvoiced stops, which become unvoiced nasals: "p : mh, t : nh, c : ngh"
So there are some similarities but also several differences.
Whilst the mutation has been explained to you, there actually is a slightly different word for nationality. In the case of Welsh, Cymraeg is language, Cymreig is nationality. In this case, the question refers specifically to a Welsh language school, "ysgol Gymraeg". A reference to a school generally in Wales, a Welsh school in the widest sense, would be 'ysgol Gymreig'. Note again that the C has changed to a G, according to Welsh rules of initial consonant mutation (research as desired).
Apologies if this is an overly technical explanation at this stage, but just for future reference. Equally, apologies if it's in any way factually inaccurate - please correct me as appropriate.
Yes. Ysgol Gymraeg is a Welsh-language school (i.e. one where the lessons in all subjects are conducted in Welsh). Ysgol Gymreig is any school which happens to be in Wales. The difference, when not appreciated, can give rise to much hilarity, as in the case of those enterprises claiming to sell "Caws Cymraeg" (Welsh-speaking cheese)...
On a side note, it was noticeable in the recent local elections that the Welsh Conservatives called themselves y Ceidwadwyr Cymreig while Welsh Labour is known as Llafur Cymru. Other parties include Democratiaid Rhyddfrydol Cymru (the Welsh Liberal Democrats) and, of course, Plaid Cymru (the Party of Wales).