My rule of thumb is that in formal Welsh everything mutates, whereas in everyday Welsh stuff like names and so on are more likely not to.
In your example (at the risk of over-thinking) you’re using an informal/coloquial verb (cwplais rather than mi/fe gwblheais) so failing to mutate is fine.
I blimmin’ love this question. Something I don’t know, which gives me an excuse to go ferreting about in grammar books and other texts, and learn something new. Thank you for asking 
So here’s what I’ve found out…
@robbruce 's rule of thumb is a good one - the Welsh Government’s style guide uses “wedi’i gofrestru gyda Thŷ’r Cwmnïau” (registered with Companies House) - so, although it’s a proper name it has mutated. But no one would worry about someone saying “gyda Tŷ’r Cwmniau” (in fact, I think that would be more normal in spoken Welsh).
However, I’m not sure it’s the same for works of art. To me, mutating something like “Traed Mewn Cyffion” (a famous Welsh novel, by Kate Roberts) just felt wrong. So I went a-Googling, and found this published quotation buried in a PhD thesis (so from a reputable writer/publisher/source in an academic context): “Sioned yw’r unig un yn Traed Mewn Cyffion a syrth…” So even in this most formal of contexts there is no mutation.
Thank you @robbruce and @sara-peacock-1 both for your amazingly quick and informative answers!
Happy to have given you a reason to go digging into some Welsh grammar too haha 
!
Probably a long shot, but is anyone aware of a Welsh translation / idiom / aphorism for the idea of ‘the tragedy of the commons’?
I believe it’s trasiedi’r tir cyffredin
edit - I’ve checked online, and this seems to confirm it https://www.cardiff.ac.uk/news/view/1310252-tragedy-of-the-commons (switch the page from English to Welsh)
Gwych, diolch!
Do names still mutate in the most formal Welsh… e.g. Annwyl Ddylan … definitely seen this in older family literature but I wasnt raised in a Welsh medium school
What does the Welsh proverb:
Nid oes ar uffern ond eisiau ei threfnu ( Hell only needs organising) … literally mean?
Is it saying that life is hellish without organising yourself… or is there a deeper meaning?
From what I’ve read, no. I believe they used to in old literature, but it’s not modern practice.
It’s amazing how newbie learners (me) arrive at their own “sense” of something they try to understand: I read it as "Hell doesn’t need, but wants to to be organised.
It’s an interpretation after my own heart 
Hi team
Looking for a little bit of help understanding what is said in a negative response sentence in Level 2, Challenge 9, if someone can help.
English
They told us that they didn’t want to explain what was happening.
Cymraeg
(sounds like)
Dwedon nhw wrthon ni na g’nhw’n moyn esponio beth oedd yn digwydd.
What on earth is the ‘gnu’ in the middle? I’m pretty sure I’ve not come across it in any of the other challenges so far…!
All wisdom hugely appreciated 
Ben
nag ynhw?
I think the ynhw is a very condensed form of the more conventional, less ‘street’ ydyn nhw.
Possibly! … Eisiau in southern dialect is closer to “need” … so I suppose northern speakers may view it very differently!
In its full “standard” form it would be nad ydyn nhw’n moyn but in speech the nad often sounds more like nag and then, as mentioned above, the ydyn nhw’n gets compacted down to just ‘y’ nhw’n so you get what sounds like nag 'y nhw’n moyn
Yes, these "nag y…’ constructions are very genuine.
I did check with a young first language colleague a while back. He confirmed that was the way his family spoke. Although, in school, his teacher suggested that he might want to be a bit more formal in an exam.
Sorry, I forgot to.add. It is probably acually “nac”, pronounced “nag”
I dug out “y hen peiriant gwnïo, achos dw’i isio gwnïo pyjamas Nadolig i teulu, ond” how on earth does one pronounce “gwnïo”? Is there a particular stress on the ï?
Apologies if I put this into something else. It’s meant to be a distinct question.
Sorry about going back to this, but couldn’t resist.
If hwfro is too modern, how about bisellu, bwthio, or Gryffithio? As precursors to hoovering.
Also a link for vacuuming geeks 
The dots over the i indicate that the i is a separate sound/syllable to the o and not an io dipthong, so the pronunciation of -ïo is ee-oh whereas the pronunciation of -io is more yo.
e.g. gwnïo = gwn-ee-oh, copïo = cop-ee-oh, but nofio = nov-yo, glanio = glan-yo
I’ll copïo this info into my brain. Diolch!