First question (almost) solved

Mehrabian, Mitsche, and Pralat showed that any $ n $-vertex graph with a good edge-labelling has at most $ n \log_2 n $ edges, and that for each $ n $ there is an $ n $-vertex graphs with a good edge-labelling having $ n \log_2 n - O(n) $ edges.

http://arxiv.org/abs/1211.2641

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