Welcome to GL&U!
Your sentence uses a double infinitive forming the perfect tense in a dependent / subordinate clause. That sounds kind of complicated, but break it down into its smaller parts and then put it back together (I did also find an English page for you that further explains some of these peculiarities).
First, recognize that a subordinate or dependent clause can change the word order around. So, you want to start off by finding the main verb of the subordinate clause (which is "sehen"), and see that it uses a helping or auxiliary verb ("habt") to form the perfect tense (just find the verbs for now and don't worry yet about if it uses ge- or not, since it's your first go-over). Next, determine who it was that did the sehen (which is "ihr"). Then add whom ihr saw (which is "mich"), and finally tag on the what you all saw me dodo (the "was" and the "tun"), which refers back to the antecedent das from the preceding sentence, which was probably something like "Tut das, was ihr..."
Now I'll try to summarize what you are specifically asking.
When forming a double infinitive, the ge- is normally dropped and the infinitive form used, although it is technically possible to use the ge- form unless you're using modals (wollen, mögen, etc.), for example.
The auxiliary verb (habt in this case) must lead the verbs in a double infinitive construction.
The accusative mich is used because sehen is the main verb doing the action, and it requires accusative.
In a double infinitive construction, the zu is normally dropped.
This construction does behave this way with any type of auxiliary verb, whether modals or haben or whatever.
These are general rules, but I hope that helps get you going. Everyone feel free to point out anything I've missed with the grammar.