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.
No harm in trying to get some more life out of it.
You'll need to patch the tube normally, and test it overnight.
Then you need to provide something to stop the tube bulging out through the tyre when inflated. Personally I'd put the largest thickest patch you have on the inside of the tyre, but others would use a piece of plastic or something else durable.
This is a back tyre so a sudden blowout is generally not as bad as a front one.
I would not ride this patched tyre a long way from civilisation.
Otherwise, a new tyre is the best answer. Your tube will be okay patched.
Finally, you need to think about what happened. Someone deliberately did this to your bike. Either park your bike somewhere more secure, or identify them and have words. Have you annoyed someone lately? I can't see how this could have been an accident.
Also, get paranoid. Check your bike every time you return... look for flats, but also look for other damage. Disconnected brakes would be most likely.