How does one claim all these forked coins from Monero's hardfork of April 2018 in a safe way? And which exchanges support these coins to dump them on?
First off, there are not 4 separate chains. There are only 2 chains: the official Monero, and a common chain shared by Monero Classic, MoneroC, Monero 0, and Monero Original. The last four are simply rebranded versions of pre-network-upgrade official Monero. Since they operate on the same blockchain, spending (receiving) coins with (say) Monero Classic will have the same effect as spending with Monero 0.
How does one claim all these forked coins from Monero's hardfork of April 2018 in a safe way?
Depends on what you call safe. Your fund on both forks will not be gone as long as (1) the forked coins' binary is not malicious, and (2) there is no replay attack when you spend your coin in one fork.
For (1), the safest is to compare the source code of any of the four "brands" with Monero's Helium Hydra (v0.11.1) source code, make sure nothing nasty is present, and compile it yourself. (This implies trust on Helium Hydra source code.) If that sounds too much work, you can try using Monero's binary for Helium Hydra, and change the seed nodes to be the forked coin's nodes. However, there's no guarantee that the old Monero binary will still work after network upgrade. It will certainly not work if any of the four "brands" undergo a network upgrade.
For (2), simply spend your coins with at least one post-fork image inside the ring. It will be automatically rejected when attacker replay it on another chain. Alternatively, spend the fork coin first with default ringsize, which is below Monero's new ringsize minimum.
If you are talking about privacy, have a look at my previous question. The two answers there basically said it all. Not claiming the forked coins is the safest. But if you have strong reasons to do so, you still have a chance to preserve part of your privacy. Make sure to spend the coins using the same ring. Note that since Monero Lithium Luna (v0.12.0) has a higher minimum/default ring size, you will be using a non-default ring size in the fork. Note also that this strategy is incompatible with the mitigation against replay attack above. I'll leave the work of suggesting a "safer" operation to a real expert.
And which exchanges support these coins to dump them on?
(Exercise left to readers: If I bought XMC from CoinEx, can I withdraw it to HitBTC's XMO deposit address?)
If you really have a bullish view on the forked coin, I recommend against claiming the fork coin. Instead, sell your Monero for BTC or USD(T), then buy XMC (aka XMO). Currently XMRUSDT is 170 while XMCUSDT is 3.66. With a price difference of 46x, you just need to sell ~2% of your Monero to get the same amount of Monero Classic, without compromising your, or other people's, privacy.
Remember, when you use the coins in both chains, you are automatically causing damage to everyone's privacy. From an economic point of view, this will devalue both coins and, ultimately, your fortune.