Why are the lights and fans in my home flickering and varying in speed?
Boy, that installer just loved his multi-wire branch circuits.  That's fine; I think they're awesome and my CH panels are full of them too.  However, there are rules you must follow in any MWBC installation.  

The #1 thing you must watch closely is neutrals.   On a normal circuit, a neutral problem just shuts off the circuit.  But on a MWBC, it creates a "lost neutral" situation where the two half-circuit voltages are higher and lower than 120V,  but add up to 240V. The high side can fry equipment and start fires, and the voltage will teeter-totter depending on the load on each leg.  So as, say, a freezer starts up, it will wibble-wobble the voltage on the other leg (and its own leg). 

I would normally blame backstab connections.  However, one of the rules of MWBCs is that neutrals must be pigtailed anywhere both hot legs are present.  You can't use a piece of equipment as the neutral path (as one normally does with receptacles in plain circuits).  So no backstabs should be involved.  

Now, if the whole house is doing this, it may have nothing to do with the MWBCs, it may be the same problem - a lost neutral - at the whole-house level.  If so, this is an emergency -- call the power company and report an outage because most likely, the problem is at the pole or service drop, and that's their bailiwick and they'll fix it for free.   You can also check this by measuring voltage between the main breaker lugs and neutral.  If it is lopsided and gets more lopsided when you turn loads on, that's it! 

Now another rule I see your installer did not follow is that whatever means of maintenance shutoff is used, it must shut off both legs of the MWBC.  Here, you use breakers, so the breakers need handle ties between them ($3, but hard to find) or change the 2 breakers to a 2-pole breaker.  That is to protect workers,  but it has a side effect: it makes sure both legs are on opposite poles.  In fact, for each of the cables, follow the red and black back to the breakers to make sure they are on opposite poles (occupying the same 2 spaces that a 2-pole breaker would use). 

If both wires were on the same pole, current from both sub-circuits would stack on the neutral wire, overloading it.