I’ve noticed that senior citizens pronounce arthritis as “arthur-itis”, as if the affliction was named after someone named Arthur. Why?
Second part: are there any rules in English for what sicknesses are referred to as “the [affliction]”? For instance, people catch “the flu” and “the measles”, but not “the cold” or “the allergies”?
I think these are the same people who say “ath-a-lete,” and it has nothing to do with age.
No rules. People sometimes decide to call a disease "the mumps,’ for instance, and it just catches on.
I don’t got that arthi-ritis, but got that durn pesky roomy-tism.
Old Southern people call diabetes “the sugar” sometimes.
And a certain subset of conditions take “teh”, as in “he caught teh gay”. :rolleyes:
There’s also the phenomenon of saying they’re going ‘to the hospital’ while some go ‘to hospital’. Whereas we never say going ‘to the school’ it’s always ‘to school’.
I thought “to hospital” was a Britishism?
Close. “To the hospital” is an Americanism. 
Inserting a vowel betwen syllables is a very common sort of change that has occurred in many languages. It’s one example of a broader phenomena called epenthesis. Here’s the Wikipedia entry on it:
My mother calls her arthritis “Arthur”, as it “Arthur is back today.” I always took it to be a euphemism–like when women talk about “Aunt Flo”–rather than the result of a liguistic proces like epenthesis.
As for arthur-itis, it’s not all about sloppy enunciation. Codgers have always had a kind of black humor about their ailments. “Going to bed with Arthur” and such are jokes we’ve all heard countless times, but we recite them anyway. It’s a small way of saying that we can laugh in the face of inescapable diseases, and we are not defeated. 
Calling it “Arthur” may be a deliberate piece of euphemism. (I’m not even certain that “euphemism” is the correct word. I don’t think that people would do it because that consider “arthritis” to be a taboo word. They would do it to be cute and have a nickname for the disease.) Adding a vowel in the middle to make it sound like “Arthuritis” very probably isn’t a euphemism. Epenthesis is a very common sort of linguistic change, and it isn’t deliberate in general.
I don’t know if you’re talking UK English, but in US English, at least, going “to the school” and going “to school” have different meanings. “To the school” means to the building and usually implies you are not a student there. “To school” means you are going to a school as a student.
I’ll try and keep that straight in future.
No, that’s not it.
“I’m going to school” means “I am currently a student” or “I am on my way to the school where I am a student in order to attend class.”
“I’m going to the school” means “I am going to travel to the physical location of the school for some reason other than to attend classes as a student.”
You never hear about anyone catching the clap these days.
However you will get “a cold” or “an allergy”. This is just the difference between the definite and indefinite article. Because you only get the flu once a year and measles once in your lifetime, you are talking about something specific. But if you get a cold, the symptoms are common enough that you can’t really pinpoint which variant you have.
To extend this, most people say they have arthritis, not “the” arthritis, since this is something that you have as a condition, rather than as an event with a finite length.
In most cases, Brits and Americans use the same usage here; Americans just stopped using it for “hospital”, for some reason.
A criminal goes to prison. Qadgop the Mercotan goes to the prison.
A student goes to school. A parent in the PTA goes to the school.
A parishoner goes to church. The guy who’s hired to repair the organ goes to the church.
See the pattern here? So far, this is the same on both sides of the pond. Likewise, in British English,
A sick person goes to hospital. A nurse goes to the hospital.
I always saw this as referring to a process rather than the place where the process takes place.
“Hospital” in England is a process, like “healing”. In America we, obviously, don’t use the word that way.