There is a related question here (with an answer similar to Andrew's --- it answers question #1).
Question #2a: NO, say take 4 points p,x,y,z such that |px|=|py|=|pz|=1 and $|xy|=|yz|=|zx|=2$.
Question #2b: YES, there is a trivial embedding into metric graph, which can be approximated arbitrary well by graph with fixed length of edges.
Question #0: Some related questions appear when one plays with definition of Alexandrov space, check also Gromov's CAT(κ)-Spaces: Construction and Concentration Link
