The straightforward answer is that it’s tradition. Baseball started as two leagues, and althought it has changed into one league in terms of organization and administration, it remains an important marketing point to differentiate between the two “leagues.” It matters to the fans, so MLB makes it so.
Another point, by the way: NL and AL teams occasionally play each other, in exhibition games or in the World Series. In each game, the home team’s rules are used, meaning that sometimes an AL team doesn’t get to use a DH, or an NL team does get to. In such cases, a player primarily kept for use as a pinch hitter will be assigned as the DH, or vice-versa, so in that sense, they’re interchangeable.
Baseball also has the oddity, compared to many sports, that ground rules are common. I don’t think there are any two ballparks with exactly the same dimensions, and there are also sometimes ground rules about things like how the stadium infrastructure affects whether a ball is fair or foul.
Plus 15 or 18 interleague games per year.
I thought a lot of the old time “purist” fans don’t like the DH.
It also leads to a big change in strategy since teams no longer have to pull a pitcher for a pinch hitter. The pitching changes have nothing to do with batting in the AL. (except maybe in rare cases)
You were doing so well until the end of number 5. Anyone can hit in that spot. If the pitcher moves to right field, the new pitcher comes in to the game in the RF’s lineup position. The original pitcher, now RF, bats in the DH’s spot, while the pitcher bats in the RF’s spot. So a team can lose the DH and a pitcher will never be in the DH’s lineup slot.
Be gentle. Longtime AL fans tend to forget how the double switch works. 
Ah yes, you are right. The relevant rule is:
Once the game pitcher is switched from the mound to a defensive position this move shall terminate the Designated Hitter role for the remainder of the game.
So it appears in this case the (new) pitcher would be batting elsewhere, since he would, by necessity, be replacing the right fielder.
Right. In fact, it doesn’t even take a double switch. The pitcher will very RARELY bat in the DH’s spot in the order, even if you lose the DH in a conventional way.
CF
3B
1B
DH
LF
RF
SS
2B
C
That’s your lineup.
Your catcher gets hurt, but your DH is your backup C. DH moves to Catcher, your new batting order is:
CF
3B
1B
C
LF
RF
SS
2B
P
The actual physical human being that is your DH can never move in the batting order, and the letters DH can never move in the batting order. But when you lose the DH, the letter P will very rarely overwrite the letter DH in the batting order. This generally only happens when a manager screws up and writes in 1B twice on his official lineup card, and DH on his own reference card. Then the P will start the game batting in place of the theoretical, but unofficial, DH on the manager’s personal lineup card.
I also want to know the answer to this. I’m unfamiliar with what the NL does.
In short, any of the substitutions or lineup-changes intended to cause the pitcher to bat less. I wouldn’t have a hope of remembering how most of them work, but I know I’ve seen some pretty complicated ones.
Well, yes, but those are two completely different games, unlike the AL and the NL.
It’s not like Major League Baseball, where the two leagues sometimes play each other, and where the winners of each meet in the final series at the end of the year. Rugby league teams only play other rugby league teams, and the same applies to union.
While individual players do switch from one code to the other (and sometimes back again), the rules are very different. And the most obvious difference, so fundamental that even someone who’s never watched the game before would probably notice, is that each code uses a different number of players. A rugby league team fields 13 players at any one time, while a rugby union team fields 15.