The best book for you is probably *A Shorter Model Theory* by Hodges.

Some comments on your question, though: First, you should be aware that the model theory of finite structures and the model theory of infinite structures have extremely different characters - so much so that finite model theory is essentially a separate subfield of logic, which is much closer to computer science and complexity theory. This can be (partially) explained by the fact that first-order logic is powerful enough to completely describe finite structures, so interesting questions in the first-order model theory of finite structures have to impose some constraints: working with fragments of first-order logic and taking complexity into account.

If you're really interested in finite model theory, you can take a look at this question, which has some references in the comments and answers. To my knowledge, the book by Ebbinghaus and Flum is the textbook on the subject which contains the most material not directly related to complexity theory (though there are probably books that I'm not aware of).

On the other hand, "ordinary" model theory is primarily concerned with infinite models, and as a result it's hard to avoid some set theory creeping in. If you're really turned off by ordinals and cardinals, I would recommend: (1) learn something about them, set theory is a beautiful subject! (2) in the mean time, concentrate on the model theory of *countably infinite* structures. This is a domain in which you get to see many of the concepts and techniques of model theory at work without any transfinite inductions in sight (except in more advanced topics: the Scott rank and Morley rank can be useful for studying countable structures, and they are both ordinal-valued).

It's also the case that many of the interesting examples in model theory come from algebra. So it's hard to achieve your requirements 2, 3, and 4. But this is why I suggested Hodges: In my experience students without a strong background in algebra and set theory find Hodges's book to be easier to read than Marker's.