Ideally, the default would be some sort of symbol that gets the idea across and takes up less space, perhaps all relevant icons on a single line instead of multiple lines going down vertically.
Then a client-side option to toggle between symbols, words, or off completely to suit the player's needs/desires. (if the toggle-off includes the uncatchable tag, a clear message at the beginning of a battle with an uncatchable Pokémon would likely be necessary)
Suggestions for the specific symbols themselves:
(For the color of the ⊘, red is the obvious option but gray or white would be less invasive and likely contrast better with the symbols beneath, making them more noticeable since that's the important symbol to recognize)
Uncatchable: a Pokeball with a ⊘ through it
Unbreedable: a Pokemon egg with a ⊘
Unbattleable: a fist like the fighting symbol with a ⊘
Untradeable: ⇄ with a ⊘, or a trade machine with a ⊘
Pokédex Immune: Pokédex with a ⊘
Has No Drops: this is the one that is trickiest to me, maybe an icon of a sack with a ⊘? Or simply a condensed icon of the word "drops" with a ⊘?
Cannot Mega Evolve: keystone with a ⊘
If anyone has better ideas or ideas for other symbols, please chime in!