I can see from the documentation that rlang::enquo() and rlang::quo() are used in different contexts. Hence, I used rlang::enysm() recently within a function declaration (see below). However, wrapped inside another SE function call, I got an unexpected error which I guess is related to lazy evaluation (and which goes away if I force(x) in f_enysm()). But it seems as I can also work around that by simply using sym(x) instead of ensym(x) since x is a string that does not convey any information about environments (as opposed to quosures). 
Is that safe?
If yes, I don't see when I should prefer ensym() over sym and the proposed use seems inconsistent with the terminology used with quo() / enquo(), expr() / enexpr() etc.
library(rlang)
f_ensym <- function(data, x, fun) {
  x <- fun(x)
  head(dplyr::arrange(data, !!x))
}
f_ensym(mtcars, "cyl", sym)
#>    mpg cyl  disp hp drat    wt  qsec vs am gear carb
#> 1 22.8   4 108.0 93 3.85 2.320 18.61  1  1    4    1
#> 2 24.4   4 146.7 62 3.69 3.190 20.00  1  0    4    2
#> 3 22.8   4 140.8 95 3.92 3.150 22.90  1  0    4    2
#> 4 32.4   4  78.7 66 4.08 2.200 19.47  1  1    4    1
#> 5 30.4   4  75.7 52 4.93 1.615 18.52  1  1    4    2
#> 6 33.9   4  71.1 65 4.22 1.835 19.90  1  1    4    1
f_sym <- function(data, x) {
  x <- sym(x)
  head(dplyr::arrange(data, !!x))
}
g <- function(data, x, fun) {
  fun(data, x)
}
g(mtcars, "cyl", f_ensym)
#> Error in fun(x): argument "fun" is missing, with no default
g(mtcars, "cyl", f_sym)
#>    mpg cyl  disp hp drat    wt  qsec vs am gear carb
#> 1 22.8   4 108.0 93 3.85 2.320 18.61  1  1    4    1
#> 2 24.4   4 146.7 62 3.69 3.190 20.00  1  0    4    2
#> 3 22.8   4 140.8 95 3.92 3.150 22.90  1  0    4    2
#> 4 32.4   4  78.7 66 4.08 2.200 19.47  1  1    4    1
#> 5 30.4   4  75.7 52 4.93 1.615 18.52  1  1    4    2
#> 6 33.9   4  71.1 65 4.22 1.835 19.90  1  1    4    1
# If I remove one level, I don't get the problematic behaviour.
f <- function(data, x, fun) {
  x <- fun(x)
  head(dplyr::arrange(data, !!x))
}
f(mtcars, "cyl", sym)
#>    mpg cyl  disp hp drat    wt  qsec vs am gear carb
#> 1 22.8   4 108.0 93 3.85 2.320 18.61  1  1    4    1
#> 2 24.4   4 146.7 62 3.69 3.190 20.00  1  0    4    2
#> 3 22.8   4 140.8 95 3.92 3.150 22.90  1  0    4    2
#> 4 32.4   4  78.7 66 4.08 2.200 19.47  1  1    4    1
#> 5 30.4   4  75.7 52 4.93 1.615 18.52  1  1    4    2
#> 6 33.9   4  71.1 65 4.22 1.835 19.90  1  1    4    1
f(mtcars, "cyl", ensym)
#>    mpg cyl  disp hp drat    wt  qsec vs am gear carb
#> 1 22.8   4 108.0 93 3.85 2.320 18.61  1  1    4    1
#> 2 24.4   4 146.7 62 3.69 3.190 20.00  1  0    4    2
#> 3 22.8   4 140.8 95 3.92 3.150 22.90  1  0    4    2
#> 4 32.4   4  78.7 66 4.08 2.200 19.47  1  1    4    1
#> 5 30.4   4  75.7 52 4.93 1.615 18.52  1  1    4    2
#> 6 33.9   4  71.1 65 4.22 1.835 19.90  1  1    4    1
Also, if I remove the intermediate function f_sym() and f_enysm() and make a direct call to f(), I don't get the probelmatic behaviour.
f <- function(data, x, fun) {
  x <- fun(x)
  head(dplyr::arrange(data, !!x))
}
f(mtcars, "cyl", sym)
#>    mpg cyl  disp hp drat    wt  qsec vs am gear carb
#> 1 22.8   4 108.0 93 3.85 2.320 18.61  1  1    4    1
#> 2 24.4   4 146.7 62 3.69 3.190 20.00  1  0    4    2
#> 3 22.8   4 140.8 95 3.92 3.150 22.90  1  0    4    2
#> 4 32.4   4  78.7 66 4.08 2.200 19.47  1  1    4    1
#> 5 30.4   4  75.7 52 4.93 1.615 18.52  1  1    4    2
#> 6 33.9   4  71.1 65 4.22 1.835 19.90  1  1    4    1
f(mtcars, "cyl", ensym)
#>    mpg cyl  disp hp drat    wt  qsec vs am gear carb
#> 1 22.8   4 108.0 93 3.85 2.320 18.61  1  1    4    1
#> 2 24.4   4 146.7 62 3.69 3.190 20.00  1  0    4    2
#> 3 22.8   4 140.8 95 3.92 3.150 22.90  1  0    4    2
#> 4 32.4   4  78.7 66 4.08 2.200 19.47  1  1    4    1
#> 5 30.4   4  75.7 52 4.93 1.615 18.52  1  1    4    2
#> 6 33.9   4  71.1 65 4.22 1.835 19.90  1  1    4    1
The ensym can take both quoted and unquoted arguments
f_ensym(mtcars, "cyl")
f_ensym(mtcars, cyl)
Based on the updated example in the OP's post, the while the sym takes the string object g is only taking three argument and the fun part of it is 'f_ensymwhich also have afun` that is not being passed into.  We can have one more argument for that
g <- function(data, x, fun, fun2) {
     fun(data, x, fun2)
  }
g(mtcars, "cyl", f_ensym, sym)
#   mpg cyl  disp hp drat    wt  qsec vs am gear carb
#1 22.8   4 108.0 93 3.85 2.320 18.61  1  1    4    1
#2 24.4   4 146.7 62 3.69 3.190 20.00  1  0    4    2
#3 22.8   4 140.8 95 3.92 3.150 22.90  1  0    4    2
#4 32.4   4  78.7 66 4.08 2.200 19.47  1  1    4    1
#5 30.4   4  75.7 52 4.93 1.615 18.52  1  1    4    2
#6 33.9   4  71.1 65 4.22 1.835 19.90  1  1    4    1
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