What if all passengers pay to check in extra baggage?
To answer the questions as asked,


We won't depart over the maximum takeoff weight. We'll do something to bring the takeoff weight down to not exceed the limit.
Generally, people fly as scheduled if at all possible, and as much as possible their bags go on the same plane that they do, and other cargo is a lower priority. There are exceptions; passengers might be denied boarding when the plane is simply too heavy with every one and every thing on board, but this is a last resort and not particularly common.
Total weight is total weight, and the more of it there is, the more limited your options become.  This is explained in more detail below.
If the people are overweight, it's probably not a factor in the numbers, because for scheduled commercial flights, most airlines use average numbers for passengers and their carry-on bags.  So it doesn't matter if they're all skinny or all heavy, they all count for the same "200 lbs".  For military charters and sports charters, actual weights are used, and some non-US carriers may use actual weights as well.


Okay, that's the what; let's discuss the "why":

There are a lot of "it depends" elements here, but in most cases, what happens is that the amount of other freight that can be carried gets reduced, and in the worst case some of the bags stay behind.  But that's probably fairly uncommon.  It's a very worst-case scenario when you can't even take all the passengers -- pretty rare, in my experience.

Your maximum takeoff weight is limited by one of three limits: 


a takeoff limit (how much the airplane can weigh to take off &amp; climb out while meeting all of the various requirements), 
an enroute limit (essentially, if you lose an engine at some/any point enroute, can you clear all of the terrain along the way to get to a suitable landing site), and 
a landing limit (your expected landing weight is no greater than the maximum allowed landing weight).


In practice, the enroute limit is pretty rare, and if the departure runway is long enough, the landing limit is usually the most restrictive.  So, most of the time, your limit boils down to: basic weight of the aircraft, plus cargo, plus passengers, plus landing fuel, can't exceed the maximum certified landing weight.  The aircraft weighs what it weighs, and a full load of passengers is essentially a given weight (most flights use an "average" weight value rather than actual passenger weights).  So the interesting variables are the landing fuel, and the cargo weight (bags plus freight).

If it's a good weather day and no alternate is required, then the required fuel for landing is one number -- let's say 5,000 lbs in a 737, assuming a fairly "best case" sort of day.  (Various operators may have slightly different numbers, but this is a high-level explanation, not way down in the details.)  But if the weather is bad, that number can go up: allowing fuel for holding, as well as for a possible diversion to an alternate.  Depending on how bad the weather is and how far away the alternate, minimum landing fuel might be 8,000-10,000 lbs for a 737, or more.

Thus... my maximum landing weight is 144,000 lbs for a 737-800.  If the airplane and the people weigh 125,000 lbs, I have 19,000 lbs left over.  If my landing gas is 5,000, then I can take 14,000 lbs of bags/cargo/whatever -- which is a lot.  If the weather is crummy at the destination and instead my min landing gas is 12,000 lbs, then I'm limited to 7,000 lbs of bags.  If today's flight had 6,000# of bags and 3,000# of cargo, it all goes in the first case, but not in the second, and it's usually easier to let the freight take the next flight than the bags -- it gets expensive to deliver bags that the airline didn't get on the same plane as the passenger!

There can also be instances where the max takeoff weight is more limiting... a high &amp; hot airport and/or short runways can limit how much weight you can take off with.  So perhaps "today" where I'm departing, the best that I can do (after considering current winds &amp; temperature and the various runways available) is a max takeoff weight of 165,000#.  If I'm burning 25,000# of gas getting where I'm going, then I'll land at 140,000# -- max landing weight (144k) is no longer the limit, max takeoff weight is.  And the computation now looks at the various weights plus min TAKEOFF fuel compared to the max takeoff weight, rather than weights plus min landing fuel compared to the max landing weight.  But the basic process is essentially the same.

Okay, disclaimer, there are various other things than can come into play that I've glossed over for the sake of simplicity -- max ZFW and improved climb and wet landing runways and plenty of other stuff.  If you're studying for your dispatcher license, this is NOT the answer to address all your what-if's.  It's just an overview.