Flexibility of word classes in Chinese
There are lots of words that can be both verb and noun. As you said， 姓 is one of them. I don't have a full list for all of them.

As for 吃， most of time it's a verb. 

In 我要吃, 吃 is a verb. To be more clear, you can put a target for it, like 我要吃饭；我要吃水果； etc.

这个吃很好 is unnatural. It should be 这个很好吃. 很好 is an adverb modifying the verb 吃. 

One case in which 吃 is used as a noun is 口吃，meaning stutter.