As mentioned in the comments, the underscore versions of dplyr verbs are now deprecated. The correct approach is to use quasiquotation.
To address your issue with select, you simply need to modify select_expr to contain multiple expressions:
## I renamed your variables to *_str because they are, well, strings.
filter_str <- "am == 1"
mutate_str <- "gear_carb = gear*carb"
select_str <- "mpg; cyl"                # Note the ;
Use rlang::parse_expr to convert these strings to unevaluated expressions:
## Notice the plural parse_exprs, which parses a list of expressions
filter_expr <- rlang::parse_expr( filter_str )
mutate_expr <- rlang::parse_expr( mutate_str )
select_expr <- rlang::parse_exprs( select_str )
Given the unevaluated expressions, we can now pass them to the dplyr verbs. Writing filter( filter_expr ) won't work because filter will look for a column named filter_expr in your data frame. Instead, we want to access the expression stored inside filter_expr. To do this we use the !! operator to let dplyr verbs know that the argument should be expanded to its contents (which is the unevaluated expressions we are interested in):
mtcars %>% filter( !!filter_expr )
#     mpg cyl  disp  hp drat    wt  qsec vs am gear carb
# 1  21.0   6 160.0 110 3.90 2.620 16.46  0  1    4    4
# 2  21.0   6 160.0 110 3.90 2.875 17.02  0  1    4    4
# 3  22.8   4 108.0  93 3.85 2.320 18.61  1  1    4    1
# 4  32.4   4  78.7  66 4.08 2.200 19.47  1  1    4    1
mtcars %>% mutate( !!mutate_expr )
#     mpg cyl  disp  hp drat    wt  qsec vs am gear carb gear_carb = gear * carb
# 1  21.0   6 160.0 110 3.90 2.620 16.46  0  1    4    4                      16
# 2  21.0   6 160.0 110 3.90 2.875 17.02  0  1    4    4                      16
# 3  22.8   4 108.0  93 3.85 2.320 18.61  1  1    4    1                       4
# 4  21.4   6 258.0 110 3.08 3.215 19.44  1  0    3    1                       3
In case of select, we have multiple expressions, which is handled by !!! instead:
mtcars %>% select( !!!select_expr )
#                      mpg cyl
# Mazda RX4           21.0   6
# Mazda RX4 Wag       21.0   6
# Datsun 710          22.8   4
P.S. It's also worth mentioning that select works directly with string vectors, without having to rlang::parse_expr() them first:
mtcars %>% select( c("mpg", "cyl") )
#                      mpg cyl
# Mazda RX4           21.0   6
# Mazda RX4 Wag       21.0   6
# Datsun 710          22.8   4