Should the orchid stem be cut back after blooming is done?

that is the usual thing to do - at least for phalenopsis.

what type of orchid do you have?

If a kitty nibbled at an orchid, are orchids toxic?
Can orchids live on a patio? In winter in California?

Depends on the orchid. Phals often rebloom if you cut the flower spike but leave the first flower nub on the spike on. A new spike usually forms.

Phal type Dens I cut when the last flower on the spike dies.

Tom