It all depends on if you mean the TM or using sketch. If you are using sketch then I think you keep it but, if you try to teach it using the TM most likely you won't get it.
Blaziken cannot learn Flamethrower by leveling up but it can learn Flamethrower via the TM 35. Blaziken's pre-evolved form Torchic can learn Flamethrower at level 43.
it can't learn flamethrower by itself but you can give it the TM flamethrower, you get that from the game corner
No, Skorupi does not learn flamethrower and can't learn it in any other way.
Ponyta can only learn flamethrower through TM's
emboar learn flamethrower at levle 43.
rapidash learns flamethrower if you get the TM flamethrower and teach it
Palkia can learn flamethrower with a TM otherwise NO
Ninetales can't learn Flamethrower unless you use TM or make your vulpix lvl 24 where it will learn flamethrower then evolve it into a ninetales.
Blaziken can only learn Flamethrower by TM, as it cannot learn it by leveling up. However, as a Torchic, it can learn Flamethrower at some level around level 50.
Blaziken does NOT learn flamethrower. but torchiccan. at level 42 i think!
Houndour will learn flamethrower at level 48
Infernape can only learn Flamethrower via TM. It cannot learn it by leveling up.