The paladin/monk multiclassing restriction is almost always overlooked or chucked, in my experience, by all but the most rules-uptight DM's. Even those uptight DM's often have lists of allowed multiclassing combos, or would sometimes allow it with a good character background or RP.
The animal companions needing to be taught tricks is something I wasn't even aware of, and have never even heard of, so I think that counts for being overlooked. I knew you had to train animals with Handle Animal if they were mundane animals, but not a Druid's or Ranger's Companion.
Needing an assistant for getting into full plate, I've never seen that used. It was just sort of assumed you could get into it on your own with enough time.
As for clerics casting with a weapon in one hand and a shield in the other, often I see this explained as a holy symbol being painted (or engraved) on the shield (or weapon) as appropriate to the faith. I really wouldn't say that holy symbols are always wooden or silver, just the example ones in the PHB: the equipment list in the PHB is not exhaustive of every item in the campaign world nor every nonmagical item an adventurer could want.
As for quick-drawing potions and/or wands, I figured any reasonably experienced adventurer (one likely to be carrying potions or wands) would have bandoleers, holsters, and other containers optimally placed to quickly draw. Heck, I've played in fantasy boffer larps where most experienced players are walking around with such a configuration so they can almost instantly pull out a potion, so I know it's quite possible, and the sort of thing that experienced adventurers come up with.
As for Encumberance, I usually see it come up at the beginning of a campaign, as everybody checks to make sure they aren't overweight when they gear up, and it might be questioned when the party is trying to really haul a lot of stuff around, but 98% of the time it gets ignored, especially the moment the party gets their first Bag of Holding and can justify carrying way more than their strength scores would allow.
In fact, I'd say capacities of bags of holding is a very overlooked rule. Most groups I've seen just assume any Bag of Holding can hold the entire party's supply of money, magic items, and similar relatively small items.
The scribing rules, depending on campaign, are either rigorously enforced (typically in the same campaigns as Paladin/Monk multiclassing), or handwaived away or greatly simplified.