I'm french so I had to try it out! Although I got 90-100% on all words, I was able to get 50+% on most words doing what I think is a strong (but totally understandable) american accent in french. That was fun!
The thing is, there are a lot of very educated French people who, when they speak English, have an extremely accentuated French accent. Typically scientists and politicians. It's like they don't make any effort whatsoever or think that the French accent is some kind of honorific mark they should cultivate.
Me, I've always thought the French accent is the worst ever and I try my best to hide it because it makes me feel retarded.
I work for a company that moved from Paris to California, so we have a large number of native French speakers who still work with us; it's an interesting dynamic.
It impresses Americans, who unconciously assume anything European is more sophisticated. Go to a high-end restaurant in the U.S. and I'll bet your maitre d' has a European accent. Or just look at the top luxury brands; why is a Mercedes somehow more sophisticated than a Lexus? French wine more sophisticated than Californian?
Most people don't realize that a language's accent is a physical thing. Your throat, nose, tongue behave differently when speaking different languages. If you're French, you can't speak English while keeping the same physical arrangement of your mouth and throat than when you speak French. So if you want to speak a language with a native accent, you have to experiment with all those parameters, and find what your voice sounds like in that language. For instance, my voice is a bit deeper in English than it is in French; this is particularly noticeable with my female friends who are speak both English and French natively - their voices tend to be very noticeably higher in French.
I've been learning Japanese, and it seems that I get best pronunciation results when my throat is in an even lower, more relaxed position than English. I intend to pick up mandarin at some point and am very much looking forward to experimenting with that, as I have no experience with strongly tonal languages.
Kids get this intuitively, but adults have a much harder time with it- it has to be taught consciously. Which is why most French people, even after living in an English speaking country for many years, retain a horrible French accent. We really ought to be teaching kids from the maternelle, much like many of our European neighbors do.
That being said, one thing I just have never been able to do are regional accents in languages I already speak. I cannot for the life of me speak with a northern French accent (my family's from the south, although I grew up in Lyon so have a mostly neutral accent), or a British accent.
Right! This is why I hate saying English words in the middle of German sentences. It's actually easier to just say them with the German accent, rather than take the effort to rearrange both my mouth and my brain.
When I speak foreign languages I sort of subconsciously imitate characters I've heard speak in movies or shows. I'm always kind of worried my Japanese will sound like a mix of Naruto and Miyazaki characters.
As for regional accents, I find that's more a matter of knowing the little differences in certain words, like extending or shortening syllables, or rather strongly altering the key vowel (eg. Québecois 'nawn' vs. Parisian 'nohn' for 'non'). As you said, has a lot to do with mouth shapes.
For a lot of regional accents, I find you can fake things by transposing certain vowels. eg. for Australian transpose all the eh's (eg. get, head) into i's (eg. git, hid) and using soft r's (git me a beah mate). Whereas in my limited experience New Zealanders have a sharper change from e's to i's - it's almost "geet me a beah". And of course knowing the local slang - adding 'mate' to the end of your sentences just sounds more Australian.
As an example, if you read Irish writing and learn the Irish vowel and consonant sounds / rules you can't help speaking with an Irish accent. E.g. Dun Laoghaire is pronounced Dun Leary with an English accent, but if you read it properly it comes out in an Irish accent.
Interesting theory but as a counterexample in Switzerland in the western side of the country people speak French natively and generally their English accent is way better than the English accent in French people. I think it has a lot to do with the culture of the country; for example the protection of the language is not considered as important in Switzerland as it is in France. In France radio stations have minimun quota for French music, something unthinkable in Switzerland.
I've heard this described as stress-timed (English) vs syllable-timed (French) languages. Stress-timed meaning the time between stressed syllables is roughly equal, whereas syllable-timed means each syllable is roughly equal length.
To native English-speaking ears it does sound kind of fancy. Kind of snobby too, though.
Nowadays it's all down to the communication and it's why I learn languages at the first place, being able to communicate and having fun learning from each other.
Unfortunately is not always the case in France, where we expect people to speak French as it should be, hopefully with more people travelling and having experiences abroad we will be more open when foreigners try to speak French.