NOTE: Because you forgot to state which <<dropdown>> macro you are using, and because I needed one to be able to test your examples I decided to use the one from this answer.
Try the following as a replacement for your first example.
<<set $gender to '', $name to ''>>\
This is a story about a <<dropdown '$gender' 'man' 'woman'>>\
named <<textbox '$name' 'Adam' 'Introduction'>>\
<<button 'Submit' 'Introduction'>>
<<if $gender is 'woman'>>
<<set $heshe to 'she'>>
<<set $capitalheshe to 'She'>>
<<set $hisher to 'her'>>
<<set $capitalhisher to 'Her'>>
<<set $himher to 'her'>>
<<set $capitalhimher to 'Her'>>
<<set $hishers to 'hers'>>
<<set $capitalhishers to 'Hers'>>
<<set $himselfherself to 'herself'>>
<<set $capitalhimselfherself to 'Herself'>>
<<set $heshe to 'he'>>
<<set $capitalheshe to 'He'>>
<<set $hisher to 'his'>>
<<set $capitalhisher to 'His'>>
<<set $himher to 'him'>>
<<set $capitalhimher to 'Him'>>
<<set $hishers to 'his'>>
<<set $capitalhishers to 'His'>>
<<set $himselfherself to 'himself'>>
<<set $capitalhimselfherself to 'Himself'>>
... there were a number of issues with your original first example:
1. You weren't initialising the $name variable, and ideally both the $gender and $name variables should be initialised within your StoryInit special passage.
2. The Label parameter of your <<button>> macro should be a String value, so it needs to be wrapped in quotes.
3. Your <<if>> macros were being executed when the Passage was first shown, and not when the Submit button was pressed. This means that the referenced $gender variable was equal to an empty String, which is why none of the related <<set>> macros were being executed.
4. You don't need to check the $gender variable for each of the related <<set>> macros, you can check it a single time then execute multiple <<set>> macros within the relevant <<if>> or <<else>> macro's body.
In relation to your second example, your first <<timed>> macro call isn't doing anything, and I'm not totally sure what you wanted it to do so I haven't made a suggests on how to fix it.