They've banned people for using modded clients in abusive ways, i.e. overriding spam/flood controls, which is reasonable. Unfortunately their official statement in response to those incidents was essentially "You can be be banned at any time for using modded or unofficial clients." They haven't actually done that to any legitimate users AFAIK, but their overreacting, user-hostile stance is worrisome, so I can see why amyjess is concerned.
It actually is. That's how slowmode works: if you send a message in a channel with slowmode, it sets a flag on your client telling it to not let you send anything for X seconds. No reason an alternative client can't just clear out that flag.
