How and why does using redirection or writing files within an if statement affect exit code?

$?

In the PARAMETERS SET BY THE SHELL section of man zshparam, or info zsh 'Parameters Set By The Shell' you’ll see that, $? is the exit status returned by the last command.

$(...)

In man zshexpn or info zsh 'Command Substitution' you’ll see:

COMMAND SUBSTITUTION

A command enclosed in parentheses preceded by a dollar sign, like
'$(...)', or quoted with grave accents, like ..., is
replaced with its standard output, with any trailing newlines deleted.

So for instance, the expression $(echo 'a string') would be replaced with the output of this command (a string<newline>) stripped of the trailing newline characters, which gives a string. But the command inside $(echo 'a string' > test.txt) would produce no output to the standard output, so the result of that expression is an empty string.

[[ ... ]]

In the man page of zshmisc or info zsh 'Conditional Expressions', you’ll see:

Conditional Expressions

A conditional expression is used with the [[ compound command to
test attributes of files and to compare strings. Each expression can
be constructed from one or more of the following unary or binary
expressions:

[…]

For compatibility, if there is a single argument that is not
syntactically significant, typically a variable, the condition is
treated as a test for whether the expression expands as a string of
non-zero length.

And now for your examples

So in the following line, the output of the command inside the $( ... ) is the string a string<newline> and the expansion of $(...) itself will therefore be a string, which is a non-zero length string, thus the [[ ... ]] evaluation will return true (0 exit status).

[[ $(echo 'a string') ]]

But in the following command, since you have a redirection, the expansion of $( ... ) will be a zero-length string, and the [[ ... ]] evaluation will return false (exit status 1, being a number other than 0).

[[ $(echo 'a string' > test.txt) ]]

Similarly, the exit status of the following command is the exit status of the curl command.

curl -so 'curl-8.2.1.tar.gz' https://curl.se/download/curl-8.2.1.tar.gz

And the manpage of curl will state under what condition it returns a non-zero (failure) exit status.

But in the next example, the exit status would depend on whether the curl command had any output (success) or didn’t have any output (failure).

[[ $(curl -so 'curl-8.2.1.tar.gz' https://curl.se/download/curl-8.2.1.tar.gz) ]]

I’ll stop here, but you can continue reading the manual, for example the meaning of (( ... )) in the ARITHMETIC EVALUATION section of man zshmisc (or info zsh 'Arithmetic Evaluation')