Learn Hardcore Japanese: ane mo onazi biyouin ni kayotte ite, itumo osyarena kamigata ni site moratte iru. - My older sister also goes to the same hair salon and always has them give her a stylish hairstyle.

It specifically means older sister (older than the speaker).
Because you say it about your own family member, you use the plain form 姉, not the polite お姉さん.
If you were talking about someone else’s older sister politely, you would say お姉さん instead.

So in this sentence, 姉 is understood as “my older sister” even though “my” is not explicitly written.

What does も do in this sentence?

も is the “also / too” particle.

Here it shows that the older sister is in addition to someone else who goes to that beauty salon. The implied context is something like:

“I go to that beauty salon. My older sister also goes to the same salon and always has her hair styled nicely.”

So 姉も = “my older sister also…”

What is 同じ doing here, and what is it “the same” as?

同じ (おなじ) means “the same” and here it modifies 美容院:

同じ美容院 = “the same beauty salon”

Same as what?
Japanese often omits information that’s clear from context. Here, 同じ probably refers to:

“the same beauty salon that I go to,” or
“the same beauty salon we were just talking about.”

So the full idea in English would be like:

“My older sister also goes to the same beauty salon (as I do)…”

Why is it 美容院に通っていて and not something like 美容院へ or 美容院で?

The verb 通う (かよう) means “to go regularly / commute / attend” and it normally takes the particle に:

学校に通う = to go to school regularly
ジムに通う = to go to the gym (on a regular basis)

So:

同じ美容院に通っていて
= “(She) goes to the same beauty salon (regularly)”

へ can also mark direction, but with 通う, に is the standard and more natural choice.
で would mark the place where an action happens (e.g., “at the salon”), which is not the main focus of 通う; the focus is “attending / going (there) regularly.”

What does 通っていて mean exactly, and how does the 〜ていて、… connection work?

通っていて is:

通う → 通って (て-form) → 通っていて (ている in て-form)

通う: to attend / go regularly
通っている: is going (regularly), attends (as a habit)

The pattern A ていて、B connects two actions or states:

A: 通っていて = “(she) goes there / attends there (regularly)”
B: いつもおしゃれな髪型にしてもらっている = “(she) always has her hair stylishly done”

So:

姉も同じ美容院に通っていて、いつもおしゃれな髪型にしてもらっている。
≈ “My older sister also goes to the same hair salon and (there) always has her hair styled nicely.”

The 〜ていて、 just links the first ongoing/habitual action (goes there) to the second habitual result (gets her hair done).

What does おしゃれな mean here, and how does it relate to 髪型?

おしゃれな is an な-adjective meaning things like:

“stylish,” “fashionable,” “trendy,” “well-styled”

髪型 (かみがた) means “hairstyle”.

So:

おしゃれな髪型 = “a stylish hairstyle” / “a fashionable hairstyle”

Together, いつもおしゃれな髪型にしてもらっている means she always has them give her a stylish hairstyle or always gets a stylish haircut.

Why is it 髪型にしてもらう? What does 髪型 + に + する mean?

The structure is:

髪型にする
literally: “to make (something) into a hairstyle”

In general, X にする means “to make (something) into X / to choose X / to set X as the result”:

部屋を きれいにする = make the room clean
髪を 短くする = make hair short / cut hair short
飲み物はコーヒー にする = I’ll have coffee (choose coffee)

Here:

(誰かが) 髪を おしゃれな髪型にする
= Someone makes (her) hair into a stylish hairstyle.

Then してもらう is added to show she receives that action as a favor (see next question). So:

おしゃれな髪型にしてもらう
= “to have someone make (your hair) into a stylish hairstyle for you”
= “to have your hair done in a stylish style (by someone)”

What does してもらう mean in this sentence?

〜てもらう is a benefactive expression:

(Person A は) (Person B に) Vてもらう
= Person A gets Person B to do V for them
= Person A receives the favor of Person B doing V.

In this sentence, it’s:

(姉は) [美容院／美容師さんから]
おしゃれな髪型にしてもらっている。

Meaning:

“(My older sister) is having (the hairstylists) make her hair into a stylish hairstyle for her,”
or more natural English:
“She always gets her hair styled nicely (by them).”

So してもらう emphasizes that:

Someone else does the styling (the salon / stylists).
The sister is the one who benefits from that action.

Why is it してもらっている instead of just してもらう?

〜ている on a verb in Japanese can show:

A continuous state, or
A habitual/repeated action.

Here, してもらっている is habitual:

してもらう = “to get (someone) to do it for you (once)”
してもらっている (with いつも) = “(regularly / habitually) get (someone) to do it for you”

So:

いつもおしゃれな髪型にしてもらっている。
= “She always has them give her a stylish hairstyle.”
= “She regularly gets her hair styled fashionably.”

The combination of いつも + 〜ている strongly gives the idea of a fixed habit.

Who is doing what in おしゃれな髪型にしてもらっている? Who is the subject, and who is the person doing the action?

Even though it’s not all written out, the roles are:

Subject (receiver of the favor): 姉 (my older sister)
Doer (stylist): the staff at the 美容院 (implied)
Action: make the hair into a stylish hairstyle

Expanded structure:

姉は (美容院の人に)
おしゃれな髪型に してもらっている。

In English:

“My older sister is (regularly) having the people at the salon make her hair into a stylish hairstyle (for her).”

So:

姉 = the one whose hair is being styled, and who benefits.
美容院の人 (or 美容師さん) = the ones who actually do the styling.
してもらっている marks that she receives that action as a favor.