You are correct, they are synonyms and can be used interchangeably in your example sentences. However this doesn't mean you can always exchange the word "but" with one of these words or vice versa.
Synonyms of "but"
A thesaurus is a great resource for finding and verifying the synonyms of a word. Looking up the word "but" on Thesaurus.com you'll find the following:
although, however, nevertheless, on the other hand, still, though, yet
Going one step deeper by looking at the synonyms of "nevertheless":
still, though, yet, nonetheless, notwithstanding, withal ....
When not to use a synonym
In the example sentences you have provided, the words are being used as conjunctions, joining the two sentences. However, some of these words can be used for other purposes. Also, depending on how grammatically correct or formal you want your writing to be, one word may be more appropriate than another.
I'd like to get to Boston however I am able.
This usage of "however" is not a conjunction and using "but", "nevertheless", or "nonetheless" instead would change the meaning of the sentence.
There are many other people more suited to the task. Nevertheless, I will do this for you.
This usage of "nevertheless" is not a conjunction. Instead it is being used as an adverb and cannot be replaced with "but". It can be replaced with "nonetheless" or "however" though, which can also be used as adverbs.