How did one know that a telegram came from the person it said on the note? I'm especially interested in how (important) politicians dealt with that problem, if it ever was one.
Also, could fake telegrams be feeded into telegraph lines?
I'm basing my answer on my knowledge of the Zimmermann telegram (the telegram from Germany to Mexico, which effectively brought the American's into WW1). Countries often shared long distance cables for telegrams and as such would use ciphers for their communications to avoid fakes and or other people reading them.
For example the Zimmerman telegram was sent using a "neutrally operated transatlantic cable". It was first handed to the US embassy in Berlin, who telegraphed it to the US embassy in Denmark and then sent to the US from there (finally being forwarded to Mexico via the German embassy in the US). The way it was intercepted was that the line went to the UK to a relay station (Porthcurno at Land's End) to be boosted for the long transmission across the Atlantic. It seems crazy that Germany was using British telegraph systems to relay their message but the line was shared between the UK, Sweden and USA (and possibly other countries that I don't know of) and as such was not cut.
As I mentioned before in order to prevent others reading the sensitive information it was sent encrypted, however the British were recording all information going through their relay station and managed to decrypt it. The ciphers used were very complex and often unique so forging them would be impossible (clearly reuse of a single pad cipher would have to be a forgery).
For more on this topic I read this book while on a plane
T. Boghardt. "The Zimmermann Telegram: Intelligence, Diplomacy, and America's Entry into World War I" Naval Institute Press (2012)
but the wikipedia entry is pretty thorough for a basic intro
https://en.wikipedia.org/wiki/Zimmermann_Telegram#British_interception
and a picture of the telegram itself...
https://upload.wikimedia.org/wikipedia/commons/8/8d/Zimmermann_Telegram.jpeg
... dealt with that problem, if it ever was one.
Yes ocassionally that was a problem. See for example this:
On 7 February 1910 the hoax was set in motion. Cole organised for an accomplice to send a telegram to HMS Dreadnought which was then moored in Portland, Dorset. The message said that the ship must be prepared for the visit of a group of princes from Abyssinia and was purportedly signed by Foreign Office Under-secretary Sir Charles Hardinge.
this worked, they apparently had no way of checking who was the real sender, so they believed it.
While I can't answer this question specifically, I highly advise reading The Victorian Internet by Tom Standage. It's a history of the telegraph and goes into how telegraph operators used the lines for their own personal use at times, as well as having their own dialect, similar to chatspeak.
