I used your synonym link. I included the words I found in parentheses. Can you check if they are appropriate, please? Thank you.
) The festival is held in Park City, Utah and represents one of the world’s premier showcases (synonym: main settings?) for new, independently produced films.
2) It was launched by actor Robert Redford to promote new film-makers and low-budget films and to help keep independent cinema alive.
3) Redford says that the festival has benefited Hollywood, since some low-budget (small-budget?) independent films coming out of the festival have earned well at the box office.
4) Movies such as the documentary March with the Penguins are “indie” projects (independent projects) that premiered (performed) at Sundance and hit the jackpot (were commercially successful) commercially.
5) One of many notable movies to premier (perform) – nearly half of which are the work of first-time film makers – is Afghan Star.
1) comma after "Utah" -- synonym you chose is a good one.
2) OK
3) "low-budget" is the common phrase; no synonym; also, "have done well" is the regular phrasing used, not "earned well"
4) Movies such as the documentary March of the Penguins are “indie” films that opened at Sundance and were commercially very successful.
5) Many notable movies have opened here, nearly half of which are the work of first-time film makers; one is Afghan Star.
1) The festival is held in Park City, Utah and represents one of the world's premier showcases (synonym: main settings?) for new, independently produced films.
The synonym "main settings" is not appropriate in this context. A more suitable replacement could be "major venues" or "prominent platforms" for new, independently produced films.
2) It was launched by actor Robert Redford to promote new filmmakers and low-budget films and to help keep independent cinema alive.
The usage of "new filmmakers" is appropriate. However, the term "low-budget films" can be replaced with "inexpensively made films" or "budget-conscious productions" to provide a similar meaning.
3) Redford says that the festival has benefited Hollywood, since some low-budget (small-budget?) independent films coming out of the festival have earned well at the box office.
The synonym "small-budget" is appropriate in this context. It accurately conveys the same meaning as "low-budget" films.
4) Movies such as the documentary March with the Penguins are "indie" projects (independent projects) that premiered (performed) at Sundance and hit the jackpot (were commercially successful) commercially.
The terms "independent projects" and "performed" are appropriate replacements. However, "hit the jackpot" can be replaced with "achieved significant commercial success" to convey a similar meaning.
5) One of many notable movies to premier (perform) – nearly half of which are the work of first-time filmmakers – is Afghan Star.
The synonym "perform" is not appropriate in this context. Instead, you can use "premier" or "screen" to maintain the meaning of showcasing a movie for the first time.