Inserting Taxonomy Images into nodes
I've done something similar by using a template for the taxonomy term based on template suggestions. Personally I prefer writing theming functions though, and example for how to do this for a taxonomy field can be found in template.php of the default Bartik theme that comes with Drupal. Look for bartik_field__taxonomy_term_reference and here is the api documentation for theme_field().

You will need to install the devel module, which will give you a dump of the node object when you are viewing a node.

You will need to either theme the entire node with specific care taken to render the taxonomy how you want from the node object, alternatively you could just theme the actual term reference field (What I first mentioned and the best way imo.) These are instructions on using the devel module.

With that you will be able to dump the array or object relating to the taxonomy term and then create a template or function theming the field to your liking, as far as I know all the information is available there it's just getting it out of the object/array can be time consuming when you are doing it for the first time.

Overall the Drupal Theming Guide covers the finer details in a lot more detail than what can be kept to a reasonable answer here.