Credits to denis for resolving issues in exporting 3D model with UV map. At first, when I tried exporting the 3D model and preview it to Unity via import the 3D model leaves it completely white at all. I put the UV map I've made in Blender and exported as .PNG file; attached onto the material in Unity and leaves a mess. Now, I took his advice and added the texture file in the textures tab under materials of the 3D model and saved. I checked at the material viewport and the UV texture appeared as part of the material.
Now, when I tried exporting the 3D model and import it onto the Unity project, the 3D model is still blank. A few seconds later after researching about this issue, I found this answer (marked in green) from the Unity Q&A section. All I have to do is to import the UV texture map (in .PNG file, exported from Blender) and attached the texture onto the material and...bam! The 3D model finally shown with the textures on it and displayed it correctly. It displayed it correctly because the 3D file is now contained with an info on UV map to display the textures accordingly.
I assumed the reason why it worked is because once you add the texture in the textures tab via Blender before saving and exporting it, it generates a UV map code and it is included in the 3D model file itself.
Here's the sample pic below. In the middle, my 3D model exported in .DAE format. Mirrored but leaves only half. No texture attached onto the material, completely blank or white. At the far left is my 3D model exported in .FBX, mirrored, 100%. This is where I took the first two attempts in exporting 3D model with a UV map displayed. It leaves black since I simply attached the UV map onto the material directly via Unity. On the right, my exported 3D model also in .FBX format. This time, the UV map displayed correctly using with Denis' simple proposed solution on UV map display and solution on about exporting with textures. The reason why there's an error on rendering the textures from that 3D model (color black) is because I forgot to add the UV mapped texture at the texture tab in Blender. I was new here on studying texture display in Blender but I'm learning it quicker. Making a 3D model has a learning curve but it is easy once I got a grasp on observing how it works.

UPDATE:
Recap on how to export UV texture mapped 3D model:
- Make a UV map via UV map editor and select specific mesh part.
- Next, once you're satisfied, save the UV by (Image > Save a Copy) and edit for colors and details in Photoshop.
- Save the new UV image map in .PNG format, attach the new image file (Image > Open Image or Alt + O) at the UV editor, and refresh it.
- Next, at the materials under a 3D object, attach the image file at the textures tab. At the mapping under textures tab, check the Coordinates if it set to UV. Check the material if a texture appears. Preview the UV mapped texture by setting the viewport to material.
- Save the 3D model and export it in .FBX.
- At Unity, add the 3D model file and the texture file. The materials folder will automatically generated.
- At the materials folder, attach the texture to the material. Add the 3D model to the scene by click and drag it. Preview it and all done!