NOTE: When referring to third-party code like the cycling link examples you mention in your question it helps if you include a link to the web-site/forum post you obtained that example from, this ensures the people answering your questions know exactly which examples you meant.
There are a number of logic/syntax errors in your example, one of which is causing the error you are seeing.
1. The (if:) macros you are using to test the current value of $count are currently mutually inclusive, this means that as the value of $count increases each previously checked condition is also true which in turn results in the current contents of the output named hook being overwritten every second. You should be using (else-if:) macros for the 2-8 value checks and an (else:) macro for the 9 value check.
2. The (stop:) macro should be in the 9 value check because there is no more content being added to the output named hook after that, you should stop your timer thread as soon as possible because they interfere with a Reader's ability to interact with the page.
The following is a modified version of your original example:
(set: $count to 0)
(set: $ThirdYears to "typing...")
(set: $count to it + 1)
(if: $count is 1)[
(replace: ?output)[<b>$uName:</b> hi.]
(else-if: $count is 2)[
(append: ?output)[<br><b>catman:</b> hey babe]
(else-if: $count is 3)[
(append: ?output)[<br><b>$uName:</b> how are you?]
(else-if: $count is 4)[
(append: ?output)[<br><b>catman:</b> not much]
(else-if: $count is 5)[
(append: ?output)[<br><b>$uName:</b> what did you do today?]
(else-if: $count is 6)[
(append: ?output)[<br><b>catman:</b> i had to help lev with his training]
(else-if: $count is 7)[
(append: ?output)[<br><b>$uName:</b> is he getting any better?]
(else-if: $count is 8)[
(append: ?output)[<br><b>catman:</b> a little bit]
<tw-link class='cyclingLink' data-cycling-texts='["i know you have a girlfriend", "why didn't you tell me?", "what kind of practices were you guys doing?"]' onclick='clickCyclingLink(this, "$ThirdYears");'>$ThirdYears</tw-link>\
(if: $ThirdYears is "i know you have a girlfriend")[(go-to: "First Passage")]
(else-if: $ThirdYears is "why didn't you tell me?")[(go-to: "Second Passage")]
(else:)[(go-to: "Third Passage")]