I was getting frustrated with copy and paste in the simulator and couldn't figure out what was going on.
Results were similar to what you describe: copy and paste would sometimes paste the same value as I had copied before, but not what I just copied to the clipboard.
I finally realized that the simulator has it's own clipboard, separate from the main OS X clipboard.
You use the touch-based copy and paste commands to paste into your application from the simulator clipboard.
You can use the standard OS X copy paste shortcuts (Command-X,C,V) to paste from the system clipboard into the simulator clipboard.
So to copy and paste from Chrome (for example), you select the text in Chrome, Command-C to copy. Then switch to the simulator, Command-V to paste into the simulator clipboard. Then in your application, you simulate a touch in the textfield to bring up the paste option, then simulate a touch to the paste button to paste the simulator clipboard contents into the app.
The menu options do the same thing as the keyboard shortcuts--copy/paste between the system clipboard and the simulator keyboard.
Now that I figured it out, I actually like it much better than if the simulator clipboard was automatically tied to the OS X clipboard.
⌘ +
C - Copies the contents of the iOS clipboard to OS X. This works for both text and images.
⌘ + ⇧ + C - Copies the selected text (or whatever else responds to the copy: selector in your app) to the iOS clipboard. This one does not appear in the menu.
⌘ + ⌃ + C - Copies a screenshot of the iOS app (without the simulator's chrome) to the OS X clipboard (Use ⌘ + S to save it to a file on the Desktop instead).
⌘ + V - Copies the contents of the OS X clipboard over to iOS, but doesn't paste it. Again, this works for both text and images.
⌘ + ⌃ + V - Pastes the OS X clipboard in the iOS app, but leaves the clipboard unchanged. This does not work with images.
I searched for an answer for this for some time. For Apple Silicon (M1) I eventually found a page where some users had tracked it down to an issue with iOS 14+. When I installed switched to a simulator iOS 13.7 as instructed (see link) pasting worked again. While this is clearly not what was going on with the original question (10 years ago!), I am pasting here in case someone else has this issue.