What order are a mesh's triangles rendered in?

Well given that the GPU is parallel there's pretty much no guarantee what it will do! I guess it's down to the hardware, the lighting path and the shader. What effect are you after? Sounds like you might need a transparent shader that does some depth buffer writing (which is not usual and still might not fix it). Not sure I get why it matters which goes first - is it that the non-transparent parts further back might need to be overlaid by the transparent parts?

Perhaps a 2 pass shader that renders only fully opaque parts in the first pass and writes to the ZBuffer followed by a second pass for only semi transparent pixels that doesn't...

The specific problem I'm having is that missiles at Y=0 are being rendered on top of the character dialogue at Y=1 (sometimes :|).

This shader renders everything in the right order, but transparent quads in this mesh fully occlude other quads in this mesh (still transparent to other meshes, which is neat, but not quite what I'm going for)

 Shader "Transparent/Z-sorted" {
 Properties {
     _TintColor ("Tint Color", Color) = (0.5,0.5,0.5,0.5)
     _MainTex ("Particle Texture", 2D) = "white" {}
     _InvFade ("Soft Particles Factor", Range(0.01,3.0)) = 1.0
 }
 
 Category {
     Tags { "Queue"="Transparent" "IgnoreProjector"="True" "RenderType"="Transparent" }
     Blend SrcAlpha OneMinusSrcAlpha
     AlphaTest Greater .01
     ColorMask RGB
     Cull Off Lighting Off ZWrite Off
     BindChannels {
         Bind "Color", color
         Bind "Vertex", vertex
         Bind "TexCoord", texcoord
     }
     
     // ---- Fragment program cards
     SubShader {
         Pass {
             ZWrite On
             ColorMask 0
         }
         Pass {
             ZWrite Off
             ColorMask RGB
             CGPROGRAM
             #pragma vertex vert
             #pragma fragment frag
             #pragma fragmentoption ARB_precision_hint_fastest
             #pragma multi_compile_particles
             
             #include "UnityCG.cginc"
 
             sampler2D _MainTex;
             fixed4 _TintColor;
             
             struct appdata_t {
                 float4 vertex : POSITION;
                 fixed4 color : COLOR;
                 float2 texcoord : TEXCOORD0;
             };
 
             struct v2f {
                 float4 vertex : POSITION;
                 fixed4 color : COLOR;
                 float2 texcoord : TEXCOORD0;
                 #ifdef SOFTPARTICLES_ON
                 float4 projPos : TEXCOORD1;
                 #endif
             };
             
             float4 _MainTex_ST;
 
             v2f vert (appdata_t v)
             {
                 v2f o;
                 o.vertex = mul(UNITY_MATRIX_MVP, v.vertex);
                 #ifdef SOFTPARTICLES_ON
                 o.projPos = ComputeScreenPos (o.vertex);
                 COMPUTE_EYEDEPTH(o.projPos.z);
                 #endif
                 o.color = v.color;
                 o.texcoord = TRANSFORM_TEX(v.texcoord,_MainTex);
                 return o;
             }
 
             sampler2D _CameraDepthTexture;
             float _InvFade;
             
             fixed4 frag (v2f i) : COLOR
             {
                 #ifdef SOFTPARTICLES_ON
                 float sceneZ = LinearEyeDepth (UNITY_SAMPLE_DEPTH(tex2Dproj(_CameraDepthTexture, UNITY_PROJ_COORD(i.projPos))));
                 float partZ = i.projPos.z;
                 float fade = saturate (_InvFade * (sceneZ-partZ));
                 i.color.a *= fade;
                 #endif
                 
                 return 2.0f * i.color * _TintColor * tex2D(_MainTex, i.texcoord);
             }
             ENDCG 
         }
     }     
     
     // ---- Dual texture cards
     SubShader {
         Pass {
             ZWrite On
             ColorMask 0
         }
         Pass {
             ZWrite Off
             ColorMask RGB
             SetTexture [_MainTex] {
                 constantColor [_TintColor]
                 combine constant * primary
             }
             SetTexture [_MainTex] {
                 combine texture * previous DOUBLE
             }
         }
     }
     
     // ---- Single texture cards (does not do color tint)
     SubShader {
         Pass {
             ZWrite On
             ColorMask 0
         }
         Pass {
             ZWrite Off
             ColorMask RGB
             SetTexture [_MainTex] {
                 combine texture * primary
             }
         }
     }
 }
 }

Well you've got a ZWrite pass there - which is going to mean the second pass might not write some of the pixels. To prove a point - just make those two ZWrite On statements Off for a moment and check whether the model then looks right - if it does we can probably figure something out with the second pass.

With both of them as ZWrite Off, it works just like the normal shader (random stuff on top). It does display transparency correctly, though.

If there's some way to "ZWrite only opaque pixels" that would be what I need for the first pass.

I'll be the first to admit I have no idea about shaders. I basically Frankenstein them together and hope for the best x)

I didn't know much until a few weeks ago - been writing this Noobs Guide To Shaders on Unity Gems and learned a lot :)

You can turn off ZWriting for individual pixels by using discard, but it's expensive.

You could try removing that first pass and allow ZWriting on the second.