When you offer food items and flowers to Diety, you are offering their spiritual essence, not their physical essence. Some people consider the essence "consumed" or "used" by Diety during the ritual time and others leave the offering out overnight or until the food just starts to turn and the flowers wilt. Obviously you would not want rotted food and flowers on a devotional altar.
Then the typiucal course of action would be to leave the food offerings somewhere outside, perhaps under a tree, for the animals to partake, leaving it with a prayer of thanksgiving. Flowers can be dried on the altar or allowed to compost outside.
When you dedicate a non-perishible item to Diety, one you plan to keep such as your crystal dog, it's assumed that you might use it for other purposes than just displaying it on an altar. An example comes from an old friend who dedicated his drum to Pan, leaves the drum by his devotional altar when not in use, but then quite regularly takes the drum out of the house to play.
Just because something is sacred and blessed does not mean it has to be kept like a nun, cloaked in spiritual bubblewrap. The godform called Hecate appreciates your offering, will be glad to "charge" this dog statue with Her energy, and then you can use or display it as you wish, always being mindful of the Divine presence within it. Maybe every dark moon or so you can place the statue on your altar in Her honor to kind of re-dedicate it if you feel better doing so. Every time you are mindful of the statue, be mindful of the Spirit within it - in that manner you continue to honor Diety.
Remember the Earth Spiritualist paths are practical paths. Many subscribe to the theory that if it feels holy and right to you, then it is absolutely holy and right. It's important not to blur this distinction with the religions of one's childhood which perhaps are more structured and filled with more prohibitions and thou-shalt-nots.
Best of luck!