Reusing "when you put [this Pokémon] onto your Bench from your hand" or "when you play [this Pokémon] from your hand to evolve 1 of your Pokémon" Poké-Powers is a matter of getting the Pokémon from play to another zone. Your options are:
(In play to hand:) Super Scoop Up, Mr. Briney's Compassion
(In play to discard:) Giant Stump, Swoop! Teleporter (followed by a recovery card to hand/deck.)
(In play to deck:) Surprise! Time Machine (failing to find), Beating Wings Pidgeot
In Magic, state-based actions are not checked while a spell or ability is resolving, but in Pokémon, Surprise! Time Machine (and Ninetales δ) does check for KOs and valid Energy/Tool attachments (e.g. Double Rainbow Energy, Solid Rage) when the Pokémon is devolved, before you carry out the rest of the card's effect.
If you have a Pokémon with a form change Poké-Power (Unown, Deoxys [ex], Ditto, Castform), you can use that Power to look at your deck, even if you know full well there aren't any Pokémon to swap for. This can be useful as your first deck search of the game to figure out what you've Prized, or if you've forgotten how many copies of a certain card you have left in your deck.
Mysterious, Root, and Claw Fossil do see play in RS-PK, but not for the reasons you might expect. They are generally not used to evolve into the Stage 1 Fossil Pokémon; that role is reserved for Strange Cave, or sometimes Holon Fossil.
Instead, Fossils are used as meat shields that don't give up Prizes when KOed. If you have an attacker like Smash Turn Dodrio, whose attack switches itself out, you can attack, send in a Fossil after doing damage, and have Dodrio sit safely on your Bench during your opponent's turn. Your opponent will either KO the Fossil with their attack on their turn, or you can discard the Fossil when it's your turn again; either way, Dodrio wil be back in the Active Spot, and you can repeat the attack, losing Fossils instead of HP on Dodrio.
This trick is also applicable if you can somehow end your turn with your (next) attacker out of the Active. Swampert ex can use Energy Recycle (which ends your turn) while hiding behind a Fossil. Weezing KOs itself when it uses Liability, and instead of sending in an actual Pokémon for your opponent to get the first strike on, you can send in Claw Fossil, which not only denies a Prize, but also threatens to take off their remaining 10 HP with Jagged Stone if they attack.
Funnily enough, Mysterious Fossil does end up being used for Evolution in hit-and-run decks. Dugtrio is a crucial tech in this strategy, as it prevents your opponent from spreading damage to Dodrio while it's on the Bench. What counters Dugtrio? Cessation Crystal. And what counters Cessation Crystal? Aerodactyl ex — which evolves from Mysterious Fossil! So these hit-and-run decks end up playing a 4-1 Aerodactyl ex line; odd enough numbers for each Stage, and even odder that they're evolving from Mysterious Fossil instead of Strange Cave, but now you know why!
The equivalent for EX Power Keepers (World Champions Pack, or WCP) was released in Japan during the DP era. By then, the rules for Fossils had changed and they gave a Prize on KO, and the text on WCP Fossils stated such. However, when playing in RS-PK, the older Fossil rule (no Prize on KO) is followed, even in Japan.
If you play a deck that uses Fossil Evolutions and Strange Cave or Holon Fossil to get them out, what are you running for Basics? The answer is, well, anything you want, and if you run only that one Basic, you are guaranteed to start with it. Some examples:
Jirachi or Baltoy in Fossil Box (Kabutops + Omastar)
Limitation Sableye in Armaldo, for Supporter locking during the early-game
Mark of Antiquity Groudon ex in Armaldo ex (an anti-Rayler deck; if you get paired against Rayler, after opening with Groudon ex, you simply refuse to put any other Pokémon into play. Mark of Antiquity prevents Rayquaza ex δ from attacking, and you have Potions and Oran Berries to heal off the tiny amounts of damage that Stantler and Minun can muster while hitting them for Weakness.)
Bad News Absol in Cradily δ (a very gimmicky knowledge check deck: if you mulligan a lot (which will happen, with only 1 Basic Pokémon in the deck) and your opponent blindly chooses to draw for each mulligan, you can hit them with Bad News to make them discard all those extra cards. Easily countered by paying attention to all the hands they reveal after each mulligan, realizing they're on this deck, and choosing NOT to draw for each mulligan.)
The ability to guarantee your starter Pokémon is made up for by the fact that you are likely to mulligan and feed your opponent extra cards, since you only have 1 to 4 Basics in your deck. (This sort of defeats the purpose of Limitation Sableye, because while you might be able to lock them out of Supporters, you're also handing them an average of 2.5 cards from your mulliganing.) Decks with Rare Candy can turbo out multiple Stage 2s with all the cards they're drawing.
There are ways for the Fossil Evolution deck to equalize: Copycat or Psychic Trace Claydol to match hand sizes with the opponent, Rocket's Admin. to bring the opponent back down to 6 cards in hand, or Bad News Absol to 5.