I. The comment is rather colloquial; the verb is implied, not stated. As the meaning is clear, the verb is not needed -- but you are correct that it still could be used.
The sentence means: "Thank God, no Yankees [were] in sight!" It could also mean: "Thank God, no Yankees [are] in sight!" just depending on the situation. The first example is more likely to be the intended meaning, however.
II. Generally, yes, following an exclamation point, one would start a new sentence. However, you will see exceptions to this. Usually such exceptions will be in writing which is not very formal in nature.
The exceptions are usually for the purpose of drawing particular attention to something, or to emphasize something. As in:
"Despite having one leg 10 centimetres shorter than the other, the boy walked 5 kilometres(!) while carrying a heavy box."
Caution should be used if you are trying something which is parenthetical like the above example. It is not usually considered acceptable in most professional writing.