I wouldn't bother about the seeds that the dog "relocated", dig the
area over again and re-sow the carrots.

One trick with small seeds such as carrots is to mix the seed with dry
sand in a jar or plastic bottle, shake the lot up to dispurse the
seeds, put a hole in the top and the just pour the sand/seed mix into
the furrow. This spaces the seed out and removes the need to back fill
shallow planted seeds (you also get to see where the seeds are planted
as the sand is (usually) a different colour to the soil).

If you are short of space try planting vegies in pots or hanging
baskets (the small "cherry" tomatoes grow well in hanging baskets and
look quite decorative).