1: It seems to be more of a naturally occurring substance than something you need to create, but it's possible.

If we do use it, exploring other continents is so far off that it might as well wait until at least one of our main PCs is a Jonin and/or old enough to mostly just do what we want.

There's other stuff that we can explore till then.

Honestly, I think that the Philosopher's stone is a bit too "western Alchemy" for the Naruto World. Alchemy appears thematically, but it's a bit more based on Chinese alchemy and Taoism: "Sennin" means Sage, but is mostly used to refer to Toaist Sages. Sage Mode is based loosely on the concept of balancing one's own Qi with that of the world in order to gain immortality(Naruto even gains it by training on a Mountain, a preferred place for Taoists in pursuit of immortality due to the belief that the world's Qi was created there before flowing down to the rest of the world) and in Naruto the key to true immortality or divinity seems to be combining the right physical and metaphysical substances within yourself(which, more often than not, includes Senjutsu) rather than creating something external to yourself.

2: I'm thinking that instead of th previously planned opening, the next mission would open with Senko alone being at Team 3's meeting place and with a scroll and saying that Jirou told her to give the scroll to Odayama with the scroll containing the details of her on-and-off assignment to the team and the "pre-selected" mission. WhetherJirou shows up in person is up to you.

A little less toe-steppy than the previous plan, should work better.

Trivia.

Senko probably refers to Jirou as "Sensei," but is using it to mean "doctor" or "professor" rather than "master." A term of respect rather than subordination to a mentor.

Since this is the Sound Village and music is important... Senko likes hard rock.

Her hobby is research. ("It's business and pleasure")

Her favorite food is teriyaki ramen, though she's also partial to wild deer(an acquired taste,) her least favorite food is plain white rice.

She most wants to fight the masters of Senjutsu who might be testing those who come seeking their knowledge.

She likes feeling the ground beneath her feet and the sun, wind, or rain on her skin.

In the event that I can't give her the Byakugan and/or it starts making more sense to go after Rinnegan instead(if she learns proof of its existence,) she probably wouldn't spend much time focusing on the advanced abilities of it.
"Wait, you're telling me that it makes learning Jutsu easier and gives you a strong affinity to all six basic types of transformation?"

"Yes but it also..."

"Who cares about what else it does? Why would I waste my time learning how to make a moon when I could spend that time improving my Senjutsu or refining my chakra?"
My logic for wanting Senko to eventually gain Lava, Scorch, Ice, Storm, and Wood Release is that they seem most "on-brand" for her. She wants to use science to better understand and connect to nature and... Well, Wood, Ice, Storm, and Scorch cover all-natural weather phenomena and biomes while Lava is the flesh and blood of the Earth.

Anyway, Igor we're waiting on you again.