Is tavla related to the French word tableau, or is that just pure coincidence?
According to Wiktionary, it originates from the German word tafel, which is in turn derived from the Latin word tabula, which is also the origin of the French word tableau. So it's a cognate but not a loanword, which is hinted at by how (in my experience) French loanwords are usually much more similar. If originated as a French loanword I imagine it would look like tablå.
Tavla = painting with a frame Målning = painting Teckning = drawing Skiss = sketch Ritning = (blue) prints
Does it absolutely have to be painted though? Could a framed work of art in charcoal be a tavla? A framed print of the Mona Lisa?
"målning" has to be painted, but "tavla" is more general, and can be drawings and photos etc. It is also used in some non-artistic words like "griffeltavla/svarta tavlan" (black board) or "stentavlor" (stone tablets)
Again, 'picture' and 'painting' are synonyms in English to the meaning of a painted work and should be accepted.
They're not synonyms, really. "Painting" is more specific; it says something about the medium that "picture" does not. Also, if you say "she gave me a picture" or "I saw a picture", I expect most American English speakers would assume you're talking about a photo, and not a painting.