For example, can a person pay to say "somewebsite.com" as opposed to "1MVY1e2vta5bA6BFEGnpqJeHUC5YaV5dsb"
There's no way the general public would use the latter.
p.s dns was an example, doesn't have to be a ".com", I meant some unique word as opposed to long randomized string.
Bitcoin addresses aren't meant to be reused as this significantly reduces privacy. This was known to Nakamoto before the initial release of Bitcoin (see section 10 of his Bitcoin paper) and his original software was specifically designed not to reuse addresses. That means any mechanism that turns easily-understandable string X into address Y should only be used once, which is not what most people want from a recognizable code word.
There have been a few described ways to work around this problem. The first is simply having the easily-understandable and reusable x be a program somewhere (e.g. myaddress.example.com) that returns a different address each time it's used. The problem with this is that it requires the receiver run a server, which most people don't want to do.
An alternative mechanism is called Stealth Addresses, which allows the receiver to create a static public key which the spender combines with a random number they select to create a secondary public key that nobody but the spender knows about but which only the receiver can spend from because it uses the same private key as before (basically). The problem with this mechanism is that the spender needs to communicate the random number they used to the receiver in order for the receiver to recognize and be able to spend the payment, which adds extra data to the block chain and costs the spender more money in transaction fees.
Both of the systems above have been tried but neither has caught on and so when people have put in work into creating decentralized persistent registries (e.g. Namecoin or Blockstack), they've not really ended up being used to facilitate payments.
Because domain names cost money and Satoshi Nakamoto wanted you to be able to lend your friend 0.00001 BTC without him have to pay a corporation an annual fee.
Because that would raise all sorts of security concerns (See DNS spoofing etc)
Because addresses are cryptographically derived from private keys and the whole Bitcoin system really rather depends on that.
