Exponentiation to multiplication to addition
Retina, 302 bytes
I'm sure this can be golfed, but at this point, I'm just glad it works. The exponentiation and multiplication sections are both very similar, but since order of operations is important, I don't know how to combine them.
y - Exponentiation
x - Multiplication
p - Addition
\d+
$*
{1`(\(\w+\)|1+)y(\(\w+\)|1+)
&gt;$0&lt;
(?&lt;=&gt;(\(\w+\)|1+)y1*)1
$1x
&gt;(\(\w+\)|1+)y
(
x&lt;
)
\((1+(x1+)*)\)(?!y)
$1
(?&lt;!1)(1+)x(\(\w+\)|1+\1)(?!1)
$2x$1
1`(\(\w+\)|1+)x1+
&gt;$0&lt;
(?&lt;=&gt;(\(\w+\)|1+)x1*)1
$1p
&gt;(\(\w+\)|1+)x
(
p&lt;
)
(?&lt;!x|y)\((1+(p1+)*)\)(?!x|y)
$1
y\((1+)p([1p]*\))
y($1$2
}`y\((1+)\)
y$1
1+
$.0

Try it online - all test cases
Test case converter
Explanation
\d+                             Convert to unary
$*
{1`(\(\w+\)|1+)y(\(\w+\)|1+)    Begin loop: Delimit current exponentiation group
&gt;$0&lt;
(?&lt;=&gt;(\(\w+\)|1+)y1*)1          Replace exponentiation with multiplication
$1x
&gt;(\(\w+\)|1+)y                  Replace garbage with parentheses
(
x&lt;
)
\((1+(x1+)*)\)(?!y)             Remove unnecessary parentheses around multiplication
$1
(?&lt;!1)(1+)x(\(\w+\)|1+\1)(?!1)  Maybe swap order of multiplicands
$2x$1
1`(\(\w+\)|1+)x1+               Delimit current multiplication group
&gt;$0&lt;
(?&lt;=&gt;(\(\w+\)|1+)x1*)1          Replace multiplication with addition
$1p
&gt;(\(\w+\)|1+)x                  Replace garbage with parentheses
(
p&lt;
)
(?&lt;!x|y)\((1+(p1+)*)\)(?!x|y)   Remove unnecessary parentheses around addition
$1
y\((1+)p([1p]*\))               Handle the 4th test case by adding if necessary
y($1$2
}`y\((1+)\)                     End of loop
y$1
1+                              Convert unary back to decimal
$.0

You may also notice that the most commonly used group is (\(\w+\)|1+). This matches an inner expression with parentheses, or an integer. I chose to use the symbols I did so that I could use \w rather than a character class. I'm not sure if it would be better to use non-word symbols and replace some lookarounds with word borders (\b).