It is kind of a jumpy animation, and could totally do with a few tweening frames, but I think if its moving down when it leans one way and up when it leans the other, it should look smooth enough in action that noone will notice. There are however two things bugging me.
1. When it leans closer, the grey bit on the front of the wing suddenly darkens, and looks pretty wrong. This is made worse by the darkest shadow in the AA sticking out like a sore thumb, but is kind wonky all around (I realize if its top lit, there is now shadow over it, but its far to intense, also, if this is set in space your worrying to much about your light source.)
2. In the middle frame, your back wing tip turns into a strip of two or three pixels, well previously it was a point. Made more glaringly obvious by the fact that those two pixels dont appear to be connected to anything at all. I would just loose them, honestly.

Also, that laser is damned saucy, as is the giant one with the ripples of energy.

*blinks* is that cheetaman?
So, I would say for the background stuff, keep the thick outlines you have on everything else with the external AA otherwise you will kill the style. That being said, I WOULD recommend making the outlines less thick as the perspective moves further from the player, and still have the thick outlines intact (seeing as they look to be 3 pixels...)
Use less contrast, and a darker pallet, possibly hue shifted slightly towards blue so it appears further back. You want your highlights to be somewhere around the shadows on the foreground stuff probably. Maybe closer to the mid tones. If your background is too vibrant, it will be distracting.

'Standard' tile sizes are basicly 8x8 / 16x16/ and 32x32.
I would go for 16x16 with the foreground stuff you have here.

I think the Klona bg's wouldnt really work with what you are doing (I also think they dont work with Klona, and kind of kill the feeling) and you should probably go with something that has a flat cartoony feel to match the style.

I think Yoshi's island might be a good starting point
but you obviously want less pastels and rounded jaggy lines, as that doesnt match the rest of your style (Yoshi's island does a great job of looking like it was drawn with crayons, yours does a great job of looking like it was drawn with a pen.)

I really like the overall ship design and pallet, there is one thing bugging me tho.
The hull of the ship seems to be at a slight angle, but the cockpit and gun turret thingy are at a sideview, so it apears the hull is leaning down well everything else is lined up with the player... that looks kind of broken, perspective wise.
I would keep the hull, rather then scrapping it, and make an angled version of the turret and cockpit, then use them all together for when the ship is moving down, and then make a new version of the hull at a side view that lines up with the cockpit and turret to use when the ship is just flying straight ahead (so if the its NOT moving, assuming this is infinitely scrolling,) followed by another version of all of them angled in the opposite direction, for when you are flying up.
If you are feeling lazy, or trying to do this in one frame, however, just make the hull sideview, as that perspective will make more sense with a constant forwards motion.

Maybe make the middle bit (the round part) a bit taller? Also you could try enlarging the nearest bit.