I’ve heard varying answers to this. Some people say only Welsh place names change. But in Level 1, Challenge 10, Aran says, “Dw i’n dod o Gorea” (Corea).
For example, is it “Dw i’n dod o…” Boston or Foston? Massachusetts or Fassachusetts? (You can tell where I live LOL!)
This is one of those stupid little questions that drives me nuts, and I think it could lead to a bad verbal habit if I don’t get it answered. 
That’s a good question and I’m not sure of the answer. I do mutate if the place has a Welsh name. So I would say, “Dw i’n dod o Fanceinion yn wreiddiol” (I come from Manchester originally), but probably wouldn’t say, “Dw i’n dod o Filton Keynes.”
But I have no idea whether that’s correct.
If your Welsh is good enough for you to be asking this question, I would say that you own the language enough for you, yourself to decide. Just do whatever comes naturally.
Aran probably doesn’t realise that he said ‘Gorea’ - mutations become so automatic that you don’t even think about them. Sometimes you hear the words coming out of your mouth (“Dwi wedi bod yn siopa yn Nhesco”) and you think “whoops - that sounded a bit odd”, but it never really matters.
Bingo! 
Can’t imagine saying ‘o Corea’, as it happens. Just one of those things. I guess some mutations are just stickier than others…
In general, English placenames do not mutate. The only exception that I’m aware of is Paris- that seems to mutate in speech- dw i’n ym Mharis. When writing, you would still use the original form.
So Dw i’n dod o Boston would remain.
However, why that is I don’t know, and of course if you don’t mutate that in speech, then people won’t notice or be bothered- they will be too interested in why you are in or going to Paris!
Hello. I find this topic very interesting because we have (we had, at least, for what I remember) the same question in breton. I don’t know now what has been decided, but I remermber that during a certain time the solution was to indicate the mutation and let the “native” form at the same time, using a slash
For example : I’m going to Kemper (or “to Kemper I am going” (depends on which element you want to insist) _“to Kempe_r” would be writen “da G/Kemper” (or maybe the contrary, da K/Gemper, but it semms to me that the mutated letter comes first. Erwann would say if he comes on this topic)
That’s for the writing. To be said, it would be with the mutated form. But I think as Robbruce : some mutated forms come “alone”, you don’t even think, and others would not, they would sound “bizarre” with their first letter mutated, and you’ll prefer tell their name in the usual form you are used to
For example “da Zublin” sounds me better than “da Dublin”, but "da Vadrid" sounds me rather “bizarre”, I would say “da Madrid”, and that’s why, I suppose, there is (was ?) this writing with the slash…
But for the countries it is more simple, because they generally begin with “Bro” (land) “Bro Iwerzhon” = Ireland, “Bro Gembre” = Wales, “Bro C’hall” = France) which makes a mutation Well : I should say “began”, because I can see that it is now changing : a lot of new speakers do not use “bro” anymore. They will say “Korea” “Iwerzhon”, (and they even say “Frañs”, which is in fact the phonic french form for France, instead of breton Bro Ch’all)
I like the using of the slash, because it can be not so obvious, for a new speaker, to recognize some names when they mutate (Zublin… Zeppelin ?..) 
You see, That’s why I find vrey interesting this question of Sororp (not at all “a stupid little question”) and I read with interest your opinions
English placenames do not mutate
Patagonia would be another exception, I suppose. Paris and Patagonia are the same in English and Welsh, of course.
When working with Welsh speaking colleagues in places like Gwlad Belg, Brwsel, Rhufain, my recollection is that they would often mutate %.
I agree with others, however, that once you’re at a certain level, you’d just say what feels right.
% the words, not themselves 
I remember when events in Catalonia were more in the news you kept seeing forms like yng Nghatalwnia in places like Golwg360.
“yng nghaerfilli” and “yng ynghaerleon” (I can never remember what the Welsh is for Caerleon) is probably easier to say than “in Caerphilly” and “in Caerleon”, where in English the caer sound seems harder to say in a Welsh way after “in” - quite often it changes to “cuh” or "Cah"and loses the “r” just to accommodate the “in”.
Mutations developed from the natural flow of speaking- yng Nghaerfili is smoother to say than yn Caerphili.
For clarity, in the same way that Caerfili is a Welsh word and Caerphilly isn’t, Catalwnia is a Welsh word, whereas Catalonia isn’t, and so therefore yng Nghatalwnia would be appropriate.
I would naturally mutate place names, whether Welsh, English o’r otherwise, especially when writing. But according to many recent discussions I’ve heard recently about this subject, it seems to be a personal preference, or whatever comes naturally to you.
So I would naturally say…
‘dw i’n mynd i Fanceinion’
‘Yng Nghaeredin mae’r haggis gorau.’
I know someone who is very proud to have learnt Welsh while living in Basingstoke! (Which sounds amazing with a mutation of course…!) 
Earlier, I was mentally composing the phrase “diolch i Google”, and wondering if it mutated to “Diolch i oogle” 
Technically, I suppose it doesn’t, but I think I’m going to call it oogle from now on.
(Mind you, that’s a bit close to "ogle " for comfort, and might get me into trouble… 
)
English names/places/things beginning with G are usually the exception and don’t lose the G in mutation because… well… oogle for example!
Also, actually, Welsh first names - you wouldn’t say “dwi’n sgwennu llythyr at Areth/Eraint/Wenno”
As has been said, your Welsh is definitely good enough for you to decide on a smooth convo by convo level - as Rob says, you’ll probably end up saying the place and mutating it without thinking. Mutations are just a flow thing most of the time (like the yng Nghaerffili example).
Corea in you’re example from Aran has it’s own place in Welsh. Whereas, I probably wouldn’t imagine someone mutating Borussia Mönchengladbach. So I think, loose rule, if the place is well known mutate or not is up to you, if it’s less well known you’d probably just get an odd look if you mutated or not…so I guess…yep, it’s up to you again
However, if someone said to me “Dwi’n byw yng Nghanterbury” i’d find that strange because there’s a Welsh name for the place (Caergaint), I may not know the Welsh name straight away, but I’d find the mutation odd to my ear.
At a slight tangent -
I quite like the Southern “air” sound for Southern Cair towns, except for Caerdydd, even though it’s Kairdiff in local English. The Southern English towns such as Caersallog/Salisbury (very much in the news at the moment) seem to follow suite.
Incidentally how is Caergwrant/Cambridge pronounced please?
I always have to stop myself (and fail quite often) from saying “Aran a Chatrin”.
Understood, but I don’t really see Catalwnia as an exonym in the same way as, say, Caersallog, given that it’s really just a Welsh spelling of Catalunya. Incidentally, when I was typing my previous, the Welsh predictive text on my phone offered me “Nghatalonia” as an option, too
Famously - “Dwi’n edrych ymlaen at êm o olff”