So I decided to do a run in FRLG without catching any Pokemon with catching defined as throwing a Pokeball to obtain a Pokemon. The only Pokemon available to me are gift Pokemon and out game trades are not allowed unless I need to evolve Pokemon if any.
I only know 4 gift Pokemon and I can't remember whether there are any more than these.
The ones I know are -
Are there any other gift Pokemon other than these?
Aside from these that you mentioned: there are still three gift pokémon before the league:
Omanyte or Kabuto (Dome or Helix fossil, Mt. Moon)
Aerodactyl (Old Ember, Pewter Museum)
Hitmonlee or Hitmonchan (Fighing dojo)
There are also Pokémon buyable in the Celadon Game Corner:
Abra, Clefairy, Dratini and Porygon in both games
And also Scyther (Fire Red only) and Pinsir (Leaf Green only)
Hope I helped!!