At the end of Melemele Island, you will end up fighting the first Kahuna. After the battle, he gives you the Tauros pager. If you hold B while riding the Tauros and run into the sturdy rocks, they will break.
For sturdy-looking rocks in the water, you will need the Sharpedo Ride Pager, which you get from Grimsley on Route 15. Just like with Tauros on land, charging with B will break sturdy-lookings rocks in the water.
There are a few places that would be good to grind at prior to this battle. This guide suggests the following locations.
Altar Of Sunne Or Moone [Recommend Level 35+]: Outside the altar where you find your game’s main legendary are patches of grass that hide Lycanroc, Skarmory and Machoke. They are all leveled at 40 or above as well.
Melemele Island’s Battle Buffet [Recommended Level 45+]: In Hau'oli City is a large mall. Up the central stairs is something called the Battle Buffet. It costs 1,200 Pokédollars to enter, but it offers the EXP reward of battles with level-50 Pokémon. The buffet can only be entered once a day (without changing the clock of your 3DS) and is only limited to 10 battle turns a ticket. That means the faster you knock out these trainers, the more experience you stand to gain. It’s a great way to accumulate EXP for the Elite Four and beyond.
Based on your team's composition, it doesn't seem like the Altar of the Moone would be a great place for you to train, as the only inherent advantageous matchup you'd have is against Lycanroc (possibly others; I don't know your Pokémon's movesets)
Based on level and your team's typing, I think the best place to grind out levels would be Vast Poni Canyon. Out of the 4 wild Pokémon you can encounter, 3 of them (Boldore, Carbink, Dugtrio) are weak to water (Primarina) and grass (Tsareena, Decidueye). The Pokémon in this area range from level 41-44, which should not be an issue for you.
To address your secondary question about Chansey, they can only be found as an SOS Pokémon. Unfortunately, the place that would have the highest level Chansey is only made available to you once you become the Pokémon League Champion (which is after you catch Lunala).
Thus, the second highest level Chansey you will find will be on Mount Hokulani as a potential SOS Pokémon from Cleffa only. However, if you take a look at Chansey's Serebii page, they do not have the potential to hold a Lucky Egg.
The only Lucky Egg in the entire game is given to you by Professor Kukui for filling your Pokédex up with at least 50 different species of Pokémon. According to this page, he only gives this to you in the postgame, as well, so it doesn't look like you have the option of getting a Lucky Egg at this point in the game.
Best Answer
I can confirm that this is not possible. I tried it myself:
First, I moved my Salazzle into a PC Box and registered it to Battle Team 1, which caused an icon to appear over it to indicate that it was registered to a Battle Team.
Then, I put Salazzle back into my party, at which point the icon indicating that it was in a Battle Team disappeared.
To verify that Salazzle was no longer registered to my Battle Team, I checked the box for Battle Team 1:
Thus, a Pokémon must be in a PC Box in order to be registered to a Battle Team, however, once in a PC Box, the same Pokémon can be registered to multiple Battle Teams, as explained on the official Pokémon site: