I know that most amateur operators simply sign at the beginning or end of a message, like
The weather is good, Kz1XXY
Would it also be customary to say that in phonetic form?
Kilo zulu 1 X-ray X-ray Yankee
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The choice of using phonetics ('alpha, bravo, charlie, …') versus plain alphabet sounds ('aye, bee, see, …') should be, and in my limited experience usually is, made based on how likely the recipient is to need them to understand.
Here are two extreme cases:
A contact made using a FM repeater at close range, among people who already know each other, does not need to use phonetics because there is little noise on the received audio and the people already know what they'll be hearing.
On the other hand, someone calling CQ on simplex using SSB either for a contest or DXing will always use phonetics, because they're trying to call anyone they can, particularly operators at the longest possible range, and so they want to maximize understandability when the recipient hears lots of noise and has never heard this particular call sign before.
If someone is doing, say, VHF FM simplex, you might hear either one. If someone is continuing a discussion rather than making the initial contact, they might not use phonetics because they're doing it for the legally required identification rather than to communicate to the other party. And so on.
In fact, hams use phonetic alphabet for their call signs (and also QTH) most of the time. The reason is simple - when you are establishing a QSO you don't know how well the other operator hears you. Maybe your signal is weak on her side, or maybe there is a QRM or QSB.
During an established QSO phonetic alphabet is used by two reasons. Firstly, you make sure that the other operator received your call sign correctly. I've been in a few situations when my suffix was recorded wrong (e.g. R2AU instead of R2AUK). Repeating your call sign is a polite way to say that it should be corrected. Secondly, all other hams may hear you as well, but you don't know how well they hear you. They may even hear only one of two hams. Repeating call signs using phonetic alphabet allows to specify who is talking and who she is talking to. Otherwise someone can think that, for instance, you are calling CQ, and intervene in the QSO.
By the way (very important!) you assumption that hams sign a message by telling a call sign is not quite correct. During the QSO hams use two call signs - the first one is who you are taling to, and the second is your call sign, always in this order. The obvious exception is when you are calling CQ.
This being said, during a long and stable QSO hams may get tired of repeating their call signs using phonetic alphabet all the time and start using regular letters, or even pronounce the call signs as if they were words.
Yes, the standard phonetic alphabet is used by radio amateurs. Of course, some people use alternatives in some circumstances - but we are required to know the standard phonetic alphabet.