Should I change the tire and tube or can it be patched? Someone deliberately pierced the rear tire of my fixed gear bike. It's a 700cX25 tire with 700cX18-23 tube inside. As you can see from the image the threads have come out. I am not sure about the severity of the damage. So what should I do of the following remedies:
1. Replace both tire and tube
2. Replace the tube and patch the tire
3. Replace the tube and leave the tire as it is
4. Patch both tire and tube
What can you afford?  If you're not really struggling financially you should replace the tire.  

If you cannot afford that (or if you need a temporary fix while you're obtaining a new tire) then you need a "boot" inside the tire.  A "boot" is some piece of flat material (typically about 5 cm x 5 cm, though size can vary based on tire size) that is flexible but still stiff enough to resist the pressure of the tube trying to poke through the tire.  You can buy purpose-made boots made from plastic or rubber, you can cut your own boots out the (thin) sidewall of an old road tire, or, popular in the US, you can fold a dollar bill up several times and use that.

The boot is used by inserting it into the tire after the tire is mounted and the tube is in place.  Pry the tire off the rim near the hole and slide the boot in between the tube and the hole.

Using a boot is not as reliable as replacing the tire, of course.  There is a danger that the boot may shift over time, especially if the tire pressure is not properly maintained.  And a boot is an even less reliable fix for a hole in the sidewall of a tire.

The tube is actually less of a worry than the tire.  If the hole is a puncture rather than a cut/slit, or it's a very short slit, then you can patch it with conventional tire patches.  If the cut is so large that a standard bike patch will not comfortably cover it (with maybe a 10mm margin) then the tube needs to be replaced.