CA$-$CE$-$EF$-$FD$-$DB$-$AB$\}$ is an undirected cycle that includes every node:

No

$\{$CA$,$ CE$,$ DC$,$ FD$,$ DB$\}$ is a spanning tree:

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AD$-$ED$-$EF $=($A $->$ D $->$ E $->$ F$)$ is a directed path from node A to F:

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AD$-$DC$-$CE$-$EF $=($A $->$ D $->$ C $->$ E $->$ F$)$ is an undirected path from node A to node F:

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$\{$AD$-$DC$-$CA$\},[$DC$-$CE$-$EF$-$FD$\},\{$DC$-$CE$-$ED$\},$ and $\{$FA$-$AD$-$DE$-$EF$\}$ are all directed cycles: