How does the NFL decide which game of the week gets the monday night slot? Is it random, or does somebody look at the schedule in the off-season and decide which games would be best? I understand they already have the schedule worked out for something like the next 10 years, does that include which games are monday night, or is it more general than that i.e. team X and Team Y play in week 1, etc…?
The NFL does NOT plan their seasons 10 years in advance. I don’t know if the Monday Night Football games are picked from a pool of games or if ABC has some pull as to which teams play. Most of the games are key divisional rivalries or key confrence rivals or are important for other reasons. Often they look good 10 months in advance but dont look so good the day before (Balt vs Den this week) and some teams won’t get a monday night game for years (decades?). When was the last time the Seahawks or Bengals were on Monday night?
They’re not random. The league (probably in some degree of consultation with ABC) makes an attempt to schedule big-name teams and high-profile matchups for Monday nights. The Monday Night schedule is set up before the season starts, and is carved in stone.
This can (and has) led to some supremely unsatisfying matchups later in the season, as two teams expected to be good turned out to be quite bad. Since Week 14 matchups between two 3-10 teams tend to draw poor ratings, ABC isn’t really pleased when this happens.
There has been talk of allowing some degree of dynamic scheduling for later-season Monday night games - mostly at ABC’s request for the reason I mentioned above. Besides simple precedent, the chief opponents of such a scheme (as I understand it) are CBS and Fox, who broadcast most of the Sunday games. They don’t like idea of ABC being able to cherry-pick prominent matchups (to a greater extent than already happens), because in so doing the games would be taken away from them. So far, nothing of this sort has happened.
As far as how far in advance MNF games are scheduled, I don’t think it’s done nearly that far. The particular details of one season’s schedule aren’t finalized until the end of the previous season (opponents are, to an extent, dependent on the previous season’s standings).
The algorithm used to determine opponents is set - this is probably what you’re thinking of that the NFL has committed itself to for several years. However, since there is some dependence on the previous season’s final standings, it’s impossible to precisely state a teams slate of opponents more than about six months before the season. In a nutshell:[list=A][]home-and-home against other three teams in your division (6 games)[]all four teams in one other division in your conference - on a three-year rotation (4 games)[]all four teams in one division in the other conference - on a four-year rotation (4 games)[]the two other (besides the one you’re playing already from case B) teams in your conference whose division standing the previous year was the same as yours (2 games)[/list=A]A rough measure of a team’s popular appeal is the number of times it appears on MNF, actually. 
Really high-profile teams may play on Monday nights as many as 3 times (Maybe 4? I haven’t bothered checking.) in a season, while the perpetual dogs (like the Bengals) may go for a decade or more between appearances.
pulls out schedule
Don’t know about the last time, but the Seahawks do have one monday night game this year, on October 14th. 
The Bengals’ last appearance on Monday Night Football was on October 19, 1992.
The Steelers beat the Bengals that night 20-0.
The Bengals’ last victory on Monday Night was October 22, 1990 when they beat Cleveland 34-13.
For a while the team had the right of rejection regarding Monday night games. Denver is notorious for bad performances on Monday night and under head coach “Red” Miller, in the 80s, the Broncos took the option of not playing on Monday night and managed to make it to the Super Bowl twice in those years (they lost rather badly in both cases).
Miller a portion of his team’s success in making it to the Super Bowl to not playing on Monday night. Since then (I think) the right of rejection has been taken from the teams and given to the scheduling god.
For a while the team had the right of rejection regarding Monday night games. Denver is notorious for bad performances on Monday night and under head coach “Red” Miller, in the 80s, the Broncos took the option of not playing on Monday night and managed to make it to the Super Bowl twice in those years (they lost rather badly in both cases).
Miller a portion of his team’s success in making it to the Super Bowl to not playing on Monday night. Since then (I think) the right of rejection has been taken from the teams and given to the scheduling god.
As already noted, there currently isn’t anything like a ten-year rotation. However, just after the AFL/NFL merger, there was a long-term schedule so that every team would play every team over a 16-year (? not sure of the number) cycle. Once that cycle ended (it also may have involved two meetings – one in each city), they went to playing opponents according to their records the previous year. Now you could theoretically go on indefinitely with two teams never meeting during the regular season.
In college football, opponents are set well in advance and it’s not unusual to take ten years before you have an open slot to play a new opponent.
The conferences were realigned this year, and scheduling algorithm changed; my understanding is that (although it was true up through last year) they’ve now eliminated the possibility two given teams not meeting indefinitely (which up through last year could only (?) happen as to an intra-conference, but outside-your-division, opponent, given that the pre-2002 algorithm provided for playing all (or at least 4 of) the teams of a (rotating yearly) division of the opposite conference each season, but had more complex rules for playing the teams in the other two divisions in your own conference).
The limit on Monday appearances is 3. Typically the Super Bowl champs and runners-up and a few other high profile teams from the previous season will get to max out their Monday Night quota.
I also believe teams are only allowed to play the ESPN Sunday night game once per year. It may be simple bad luck, but ESPN seems to have a habit of picking bad matchups for its games. This is probably because Fox & CBS get first dibs on non-Monday matchups, the best Sunday game is hardly ever the ESPN night game. ESPN had the misfortunate of having to showcase the Bengals during Week 3.