Should I use %$% instead of %>%?

No, you shouldn’t use %$% routinely. It is like using the with() function, i.e. it exposes the component parts of the LHS when evaluating the RHS. But it only works when the value on the left has names like a list or dataframe, so you can’t always use it. For example,

library(magrittr)
x <- 1:10
x %>% mean()
#> [1] 5.5
x %$% mean()
#> Error in eval(substitute(expr), data, enclos = parent.frame()): numeric 'envir' arg not of length one

^{Created on 2022-02-06 by the reprex package (v2.0.1.9000)}

You’d get a similar error with x %$% mean(.).

Even when the LHS has names, it doesn’t automatically put the . argument in the first position. For example,

mtcars %>% nrow()
#> [1] 32
mtcars %$% nrow()
#> Error in nrow(): argument "x" is missing, with no default

^{Created on 2022-02-06 by the reprex package (v2.0.1.9000)}

In this case mtcars %$% nrow(.) would work, because mtcars has names.

Your example involving .$hp and .$mpg is illustrating one of the oddities of magrittr pipes. Because the . is only used in expressions, not alone as an argument, it is passed as the first argument as well as being passed in those expressions. You can avoid this using braces, e.g.

mtcars %>% {plot(.$hp, .$mpg)}