1) The answer depends on the type of logic under consideration. Here are some examples: a sentence S is logically valid iff:

• Propositional: S is a tautology.
• First-order: S is true in all models, under all variable assignments.
  
• Modal propositional: S is true in all models, at all possible worlds.
  

Note that logical invalidity can be defined in terms of logical validity as follows: sentence S is logically invalid just in case ~S is logically valid.

2-3) The answer to both questions is 'no'. Take a principle S and a logic L that validates it. Get rid of S and all other principles that are logically dependent on it, obtaining a logic L'. This L' will still be a logic but it won't be able to determine the validity or invalidity of S. That means, there can be no set of principles that all systems must obey. Thus, no statement will be (in)valid for all logical systems.