..About wether or not flatbows existed in the middle ages, they did.
There are some founds of more or less complete bows, and lots of fragments indicating that a "flatbowish" design were all of a sudden very popular in Norway in the later middle age period. (Starting from 1100+)
Theese bows are often referred to as "tvividr" -or "two-wood" bows, as they were laminated using pine compression wood for the belly, and mostly birch for the back. As the pine compression wood is the surviving parts of theese bows, and the wood for the back in most cases have not survived, one might guess if there has been other woods on the back too..
This type of bow were often as long as the earlier typical yew or wytch elm viking warbows, but were now often designed with more flat limbs and a narrow handle. (some were still designed using the old typical longbow design) The estimates of theese bows indicate that we're still talking heavy bows.
Pine compression wood is quite easy to find up here, and combined with a wood that are good in tension the result is just as good or better than yew.. Theese bows were most probably a result of the vikings learning about this from the Saami bowyers.
Ivar Malde has done lots of good research on theese bows, and made several replicas.