Can Unity bake lighting/mapping from a projector?

This can actually be done with unity pro, and a bit of shader knowhow. What you do is you write a custom vertex shader that unwraps your mesh according to its UV structure and transforms it into screen space. Then you perform your regular texture projection in the same shader. Assign that shader to a material, assign that material to a projector, put that projector onto a layer, and cull the main camera from rendering that layer, then create a second camera that renders only that projector layer and tell it to render its output to a RenderTexture. You will then have a render texture with your projector sprite projected onto its UVs.

Should be relatively straight forward for anyone who knows their vertex / fragment shaders. Here is a basic one that I came up with. I would wager a guess that this method could be easily translated to light cookies as well.

 Shader "Custom/UnwrapShader" {
     Properties {
         _ProjectedTexture ("Projected Texture", 2D) = "" { TexGen ObjectLinear }
     }
     SubShader {
         Tags { "Queue" = "Transparent" }
         Pass {
             Cull Off
             AlphaTest Greater 0
             Blend SrcAlpha OneMinusSrcAlpha
             
             CGPROGRAM
 
             #pragma vertex vert_main
             #pragma fragment frag_main
             #include "UnityCG.cginc"
         
             sampler2D _ProjectedTexture;
             float4x4 _Projector;
         
             struct Output_Struct {
                 float4 position : SV_POSITION;
                 float4 texcoord : TEXCOORD0;
             };    
             
             Output_Struct vert_main (appdata_tan v) {
                 Output_Struct OUT;
 
                 OUT.position.xy = v.texcoord.xy * 2.0 - 1.0;
                 OUT.position.z = 0;
                 OUT.position.w = v.vertex.w;
                 
                 OUT.texcoord = mul(_Projector, v.vertex);
                 return OUT;
             }
             
             float4 frag_main (Output_Struct OUT) : COLOR {
                 half4 tex = tex2Dproj(_ProjectedTexture, UNITY_PROJ_COORD(OUT.texcoord)) * half4(1,0,0,1);
                 return tex;
             }
     
             ENDCG
         }
     }
 }

No, thats not possible.

Projectors only exist at realtime.

Lightmap baking is restricted to static lights and as far as I know does currently not even support light cookies