We show that as long as there are 17+ strings, there's a move that doesn't immediately lose, but once we're down to 16 strings, the next move loses. Each move removes a string, so Alice will lose because she cuts when there are an even number of strings (28 strings, 26 strings, ...., 16 strings).
Treat the ceiling as a single node, making the construction a graph with 17 nodes and 28 edges. As long the graph has at least as many edges as nodes (17), it cannot be a tree, and so must contain a cycle. Cutting any edge in that cycle cannot cause anything to fall because any path to the ceiling that used that edge has a "detour" where the rest of the cycle was used in place of that edge.
But, once we're down to 16 edges, the graph (if still connected) must be a tree, and removing an edge will disconnect it, so the next player loses.