This one is obvious. China has a difficult relationship with not one, but
two three nuclear armed countries, two of which have far-away territories: USSR/Russia (with whom they had military conflict around Amur river); USA (with whom they had military conflict in Korea); and less-importantly, India.
Of those 3 countries, USA definitely requires an ICBM, and I suspect parts of Russia/USSR in the East as well.
First of all, the theoretical need: France during cold war faced a threat from Russia. That's 2-6 thousand miles. That's lower than IRBM's range, and requires an ICBM.
Second, in practice, France has no IRBMs anymore, and never had ICBMs, relying on SLBMs:
France now only deploys submarine-launched ballistic missiles, with all land based IRBMs decommissioned in September 1996. The French Air Force and French Navy retain aircraft-carried nuclear-tipped cruise missiles (SCALP-EG) to fulfill the pre-strategic role (tactical-sized weapons used as "ultimate warning" before launching an all-out strategic strike).
M45 SLBM (progressively retired); M51 SLBM (three variants : M51.1 from 2010; M51.2 from 2015; M51.3 projected from 2025 onwards);
Inactive: S2 IRBM; S3 IRBM; M4 SLBM
Jerico III entered service in 2011, after Pakistan went nuclear. Let's also not forget that - despite the irony that Jericho II predecessor was presumed to be a joint development with Shah Iran, Iran turned into a bitter existential enemy with a nuclear weapons program after 1979.
Distance from Israel to Pakistan 3,292 km = 2,046 miles. That well exceeds the range of previous generation Jericho II IRBM which was under 1500km - which isn't enough to reach Pakistan (3292 km), or even most parts of Iran (1789 km).
The UK has submarine-launched Trident II missiles that have a listed range of about 7,500 miles (and speculation that its even higher). Considering that the submarines could be more or less anywhere it is effectively an ICBM. I don't know why the UK doesn't have any land-launched ICBMs though.
As for why do countries have ICBMs? For the same reason countries do all their nuclear missile activities: the threat. In the cold war the concept of Mutually Assured Destruction (MAD) arose. It basically said that you can't nuke me because Ill nuke you back and we'll both die. Well I can't nuke you back if I can't reach you, so ICBMs were born. In addition to the countries you listed the other two major players who have ICBMs are the US and Russia.
Now who do these missiles get pointed at? Well, anyone the missile owner wants really. Flight plans can be modified. Realistically you end up with an east-vs-west. Exact details of ICMBs are obviously classified, but China surely has some pointed at the US, Europe, and maybe even Russia. France probably points to the Asian sphere, as does Israel (though Israel probably also points to Iran and some other middle east enemies).