Hi! I’ve got some answers to your questions (though there might be spoilers):
Boat Ticket: You get it in Violet City after earning all 8 Johto badges.
Apricorn for upgrading the Water Bike: It’s located near the Safari Zone.
Pikachu puzzle: You’re right about that one, but the Pikachu needs to know the move Surf.
Ilex Forest tree: You can’t do anything with it yet until you unlock Kanto.
Item to cut bushes: You need the bike upgrade. Once you have it, surf to Route 27.
Ruins of Alph: You need the Rare Orb item and a Psychic-type Pokémon in your party. Once you have them, go to the ruins and interact with the lone tree at the top-left of the map.