No, they don't get destroyed by default, but if the enemy is grouped together in one direction then the missiles don't spread far enough and hit each other. If enemies are spread all around you it is better, missiles don't get tangled together.
Hitpoints for parent MultiRocket looks ok (is unchanged default 10,000) - that is far more hp than the potential damage (30 warhead x 10 dmg = 300 dmg). Even if they all hit each other it is still only 4 missiles per launch, or 300 x 4 = 1,200 dmg. Each warhead has very low hp though (default 10 for MIRV_WH). Perhaps this is the problem. Let's see what happens if I jack this up to 10,000 as well.
Edit:
Ya, that did the trick. I forgot I was modding LRM1 and pointing to MIRV3, needed to increase the hp on the MIRV3 missile and warhead (I don't know why the hp for these is default 10, very very low). I guess this solves the problem, you have your MultiRocket - MIRVs now