There really is no evidence that an ice age happened. On the other hand, many cultures have ancient stories of floods.
There really is no evidence for a universal flood. On the other hand, many cultures have ancient stories of glaciation.
There’s also no evidence for Noah’s existence, apart from the tale in Genesis, which is itself obviously implausible at best, and it shares with most other flood stories (and other global disaster stories) from around the world the characteristic of having the rest of the Earth’s living things die.
The manure (which would have wound up 100s of tons) could have been shoveled over the side during the voyage. But the 37,000 species he and his family would have to have taken aboard would have required both a larger boat, and a bunch more boats tied behind to hold the food supply for the animals.