Is there a search syntax for Limewire, such as with AND, OR, NOT, +, -, parentheses, etc?

No. LW's search function is fairly basic. There's also characters the search function doesn't recognise such as parenthesis, etc. " , ' - etc are all considered as blank spaces or a missing part of the complete search. But that can be handy for searching such as adding the file type at the end of the search. eg: my movie - .avi