This indeed wouldn’t be possible to do from GUI, because that one provides access only to the basic setup parameters available on the original Memrise site. More advanced modifications are supposed to be done directly in code after creating the base for the template, so you were on the right track.
If you want to add audio on the front of the Card, which already has another Field as the question, you can copy the code for the audio block from the back of the Card to the front. If your audio Field is called “audio” (capitalization matters!), the block should be structured like this:
{{#audio}}
<div class="mem-field no-alts">
<label>audio</label>
<h4>{{audio}}</h4>
</div>
{{/audio}}
So you can copy it from here as well and adjust the “audio” field Name (in four places!) to match yours if necessary.
As for the pasting, you can put it right before <div class="mem-typing"> line on the front side, for example. This will insert the small audio button on the left side below the question image and above the typing element/multiple-choice buttons.
Alternatively, you can insert it anywhere else, depending on your preferences. The code in the block and the associated styles can also be adjusted to appear in any desired way.
Do I understand correctly that the question Field in all three Card Types is the “question diagram” (along with the audio, if the above is implemented)?
I’m not sure what problem this refers to. By default, the backside of all Card Types should display all the Fields, including the “answer image”. The Front side only displays the question Field and, for multiple-choice, the answer Fields (as buttons). So, if I understand your setup correctly, the “answer image” Field should be showing on the front only on the second Card Type, where a correct “answer image” is supposed to be selected from several options.
You can change the order of the Card Types after a Note Type has been created. In the same Anki window where you modify a Card Type code, there is an “Options” dropdown menu (actually, there are two – I am referring to the first one, which is selected right after the Card Type dropdown menu), where you can find the “Reposition Card Type” feature.
But! The order in which the Cards will be introduced in learning is not directly related to the order of the respective Card Types. If you want to achieve a specific ordering, you will probably have to adjust it in Anki Browser after all the Cards have been generated, either way.
I would assume that typing Cards are already scheduled for you first, because they are generated as soon as Notes are added, while the multiple-choice Cards only appear after “Fill Choices” is applied. However, the default order will most likely put all 144 typing Cards first, then all 144 multiple-choice Cards of the first Type, and then 144 multiple-choice Cards of the second Type. You can check this in Anki Browser: with the Cards/Notes toggle being switched to “Cards”, the “Due” column should indicate the ordinal number for each Card (in the format “New #number”). To change the order of Cards, you can use the “Reposition” function: select the Cards → right click → Reposition or Ctrl+Shift+S (more info in this manual page: Browsing - Anki Manual, if it helps)

