Do I have to challenge monsters in order?

Yes

If an edge is used multiple times, will its weight be added multiple times to xor of all edges in walk or only once?

If an edge is included multiple times in the walk, its weight contributes to the XOR as many times as it is included in the walk. / walk に同じ辺が複数回含まれる場合、その辺の重みはwalkに含まれる回数だけXORに寄与します。