I do not see any reason to show the offer letter. If the school you are negotiating with does not believe you, then that is not a good sign. As for mentioning the other university, it depends on if you can pitch it as a selling point that makes the other offer better. While you should always tell the truth, you can be somewhat selective and play both sides of the coin.
For example, if you get offers from both Stanford and Harvard and you grew up on in the Bay area, but are currently doing a post doc at MIT, you could tell Stanford that you have an offer from Harvard and that you are really happy in Boston, so they will have to beat Harvard's offer. Similarly, you can tell Harvard that you have an offer from Stanford and that your family is in the area so Harvard will have to beat Stanford's offer.
Alternatively, if you have offers from both Stanford and Bunker Hill Community College and you grew up, were educated, and are currently in the Bay area, you would not want to mention the name (bunker Hill) when talking to Stanford, unless you can explain to Stanford what benefits Bunker Hill uniquely provides. You probably want to specifically mention Stanford when talking to Bunker Hill.
If, however, you grew up, were educated, and are currently in the Boston area, then you are in a gray area and may want to mention Bunker Hill to Stanford. Given the difference in the ranking of the two universities, it might be a hard sell to convince Stanford that the Bunker Hill offer is really competitive to the Stanford offer.