Flags present a challenge to any ship modeler. Maybe the best approach to it is to start by getting a thorough understanding of the real thing.
Real flags, especially during the sailing ship era, were often extremely large. In order for them to be practical in terms of handling, they had to be made of relatively lightweight material. (I suspect another consideration had to do with the ship's navigation. Some of the flags flown by those old ships were as big as their topgallants; if a huge flag got snarled up in the rigging in just the wrong way it could really mess things up.) The surviving examples indicate that flags often were made of some tough material with the texture of coarsely-woven gauze - so coarse that the wind could literally blow through it. The old master marine artists often made their flags look downright translucent. (Here's an example: http://www.artforbeauty.com/sailpaintings/Resolution3582.html .)
The museum where I used to work had a big British blue ensign that had belonged to the liner Queen Elizabeth. The flag itself was gone by the time I got there (the file indicated "destroyed by rodents" - a thoroughly embarrassing confession for any museum to make), but the file photos established that it had been made of a material that most people would describe as "netting." The openings in the mesh must have been almost an inch wide.
The Wasa's flags (I'm basing this on the box art) were, with the notable exception of the big ensign on the stern ensign staff, pretty simple in design: yellow crosses on blue backgrounds. I imagine the real ones were made by stitching together pieces of blue and yellow fabric. The big ensign, with the heraldic figures on it, is another matter. The lions and other devices probably were embroidered separately and stitched onto the background fabric. (An interesting question: would the lions on the other side of the flag be facing right or left? The rules of heraldry are, I believe, quite specific about that - but if the lions were facing to the left as seen from both sides of the flag, and the basic fabric of the flag was indeed translucent, the result would look pretty odd - especially if the sun was behind the flag.)
Fortunately, for the modeler working on 1/144 scale much of this isn't directly relevant. My own usual approach to the problem is to paint the flags with diluted acrylic artist colors on thin white tissue paper or drafting vellum, with the paper taped down over a pattern (which is covered with Scotch tape to prevent the paint from soaking through the flag and sticking to the pattern). I deliberately keep the colors a bit pale, to give at least an impression of that transucent look. When the paint's dry I cut the flag out and take a look at the reverse side. If the design is fairly simple the paint usually will have soaked through enough that I only have to touch up a few spots. In a U.S. flag, I have to paint the stars on both sides. My target is to make the flag, and the paint, thin enough that light can shine through it - at least where there's no additional layer of fabric. (Take a look at the first shot in the movie "Saving Private Ryan." It shows an American flag with the sun behind it. The stripes and the blue field are translucent, but the stars look almost black. That's because they were sewn onto the flag separately - and they're in silhouette.)
This process is easier than a lot of people seem to think. Believe me, painting those flags would be a hl of a lot eaiser than painting the "carvings" on the Wasa. And if you don't get your flag right the first time you can throw it away and start over.
But I've never tackled anything as intricate as those Swedish lions. I'm not sure I could paint them acceptably - though using the Airfix flag as the pattern would certainly help.
One tip. Some people use colored marking pens to make flags. If you do that, make sure the pen is clearly marked "Permanent." Most of them aren't, and their ink fades almost beyond recognition in a few weeks.
The paper on which the manufacturers print flags is indeed ludicrously thick for the purpose - especially when folded over double. Maybe a partial solution (which I haven't tried, but I don't see why it wouldn't work) would be to make a color copy of the flagsheet with an inkjet or laser printer onto a thinner paper. (I do know that my Epson RX580 works perfectly well with drafting vellum. I suspect it would even print on tissue paper, if the tissue were taped down to a heavier backing sheet.) Maybe the copies could be cut in half and painted on their backs to match the fronts. Even if they were doubled over, they'd be a lot thinner than the ones in the kit.
Sailing ship kit designers do, thank goodness, seem to have abandoned a remarkably silly tradition that used to be common in the business: making flags with "ripples" drawn in perspective. I remember in particular the Revell Santa Maria, whose flag sheet featured exquisitely intricate Spanish Renaissance designs printed in great detail on what amounted to "pictures" of flags fluttering in a "breeze." The person who drew those designs (twice - once for each side of the foldover flag) was a real artist. But didn't it occur to anybody that a two-dimensional picture of a rippling flag attached to a three-dimensional ship model would look utterly ridiculous? And that anybody who's capable of dressing himself is also capable of putting genuine, three-dimensional ripples in a miniature flag in a few seconds? (By coincidence, Revell Europe has just reissued that kit - complete with the original flags: http://www.revell.de/3d-view/galery.php?objID=05405&lang=en . See what I mean?)
Just how the finished flag should be shaped is worth thinking about. I personally tend to like models with furled sails or bare spars. (That's a personal opinion - and subject to change.) On a model like that the flags, to my eye at least, look best drooping - especially if the model is a full-hull one mounted on pedestals. Einemeink's Wasa, on the other hand, seems to be experiencing a moderate wind, which is filling the sails. That wind obviously would have a similar effect on the flags. Given that the yards are squared, the wind must be coming from dead astern or nearly so; the flags therefore would be blowing almost straight ahead. (A seventeenth-century ship couldn't move fast enough to make the flags stream out behind. Note that in that Van de Velde painting, the wind is blowing from the port side and the flags are streaming out to starboard.)
Some time ago I had to do a drawing project that involved showing flags blowing in a breeze, and I quickly found out that drawing a flag in that configuration isn't as easy as it looks. So I picked a breezy, sunny day and took my digital camera, with the shutter speed set high, over to the campus library, which has a nice big flag on the pole in front of it. I got some curious looks from people who couldn't imagine why I was snapping dozens of pictures of that flag waving in the breeze, but when I was done I had a useful collection of pictures showing the configurations real flags get into. (Some of those shapes were a little surprising.)
Too long as usual, but I hope it helps a little. That fine model really deserves some nice flags. Good luck.
Later edit: I must have been typing this post when warshipguy's and einemeink's went up. Looks like einemeink thought of the same idea I did.