These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Considering that the game group that made Toejam and Earl's soundtrack later made that of Kid Chameleon...
Demonic Spiders: The Lawnmower Men, The Chicken Infantry, The Ice Cream Truck and the Mailbox Monsters in the original. Lawnmower Men move very fast on grass, are difficult to outsmart, and do extremely high damage. The Chicken Infantry fire harmful tomatoes from their cannon, with longer range than your own. The Ice Cream Truck is extremely fast, can teleport, and is almost impossible to avoid without item assistance, and the Mailbox Monster, the worst enemy in the game, disguises itself as a helpful mailbox from where you can order presents, and attacks you if you get too close.
The Boogie Men are the worst. They're faster than you, they're invisible (aside from the occasional shadow appearing for a split-second and only when they move), they're everywhere on the higher levels, and exist on some lower levels unlike the other demonic spiders — combined with the lack of Mercy Invincibility, the sound of "BOOGIE BOOGIE BOOGIE" is almost inevitably followed by the sound of a thrown controller. The Boogie Men's preferred terrain is sand, which TJ & E move slowly on, unless they are walking along an edge. Being on the edge means one "BOOGIE BOOGIE BOOGIE" will knock you down to the previous level, but that may be preferable to being caught in a Cycle of Hurting. Unless you fall into an area next to two Boogie Men.
They retain their Demonic Spider status in the sequel. On their own, they're just a nuisance, but the game will eventually hurl large groups of 3 or 4 (or even more) at you, where they can quickly surround you and wreck your health bar. And the fact that they're invisible makes it harder for the "jump into the air and carpet bomb the area with jars" trick to work.
But the Lawnmower Men are far worse. Their sheer speed combined with the aforementioned lack of Mercy Invincibility basically translates running into one of these guys into a One-Hit Kill. As if that wasn't bad enough, they'll still chase you after you die, so unless you have a present that enables a speedy getaway, once your after-death invincibility wears off, chances are you'll just die again. This is enough to cause said thrown controllers to go through televisions. All of a sudden, the pits that take you to the previous level, normally an extremely annoying thing to happen, become your best friend. The only mitigating factor is they move a little slower on road.
More on the mailbox monsters: they show their eyes at random intervals so if you wait around them long enough, they'll eventually reveal themselves. This actually makes them worse because if you camp out for 30 seconds, you might just foolishly assume that there's no way that can be a mailbox monster.
Dogs in Panic on Funkotron. They move quickly and are harder to hit than most enemies, but a single dog by itself is more of an annoyance. However, the game also likes to stick more than one dog with a dog lady, where they'll quickly tear you apart if you aren't quick on the draw or have Super Jars or a Funk-Vac handy. Finding just a single dog lady is even worse, because it usually means that the dogs are lying in wait for you to capture her before they jump out and ambush you.
The naked men in cardboard boxes have more health than any other enemy, and have an annoying tendency to hide in their boxes (which makes them invincible) and walk around to ram into you repeatedly for massive damage if you hit them. The general strategy is to jump and hurl jars as fast as you can and hope that you capture them in one go before they retreat into the box.
Fridge Logic: You can steal presents from Santa, presents that were presumably intended for human children. These presents are totally random and do include the "Total Bummer" packages that instantly kill you. So... Santa is trying to straight-up murder naughty children now? Whatever happened to lumps of coal? On the plus side, thinking that Santa is secretly a child killer DOES alleviate most of the guilt you might have felt for stealing from him.
Perhaps he saw Toe Jam and Earl ahead of time in that instance and put the bad package on top knowing it would be stolen. Well played, Santa.
Game Breaker: One of Trixie's level-long power-ups in Panic On Funkotron is unlimited Funk-Vacs, turning any encounter with Earthlings on that level into a complete curb-stomping.
Goddamn Bats: Wahini, the hula woman, has a tendency to cause paralysis in the form of a contagious hula dance. Cupids can mess up the controls with their arrows, and stay out of range from normal attacks.
Tornados are even worse. They won't kill you (so they're proper Goddamn Bats), but they do pick you up, carry you around randomly for a while, and more than likely drop you off over open space, so you fall back down to the previous level.
The Moleman/Dogman. He doesn't hurt you, but he will remove A LOT of presents from you, and moves very quickly. Oh, and he mostly appears in quicksand levels where your moving speed is drastically lowered.
Tourists in Panic on Funkotron can stun you with their cameras (and deal damage at the same time) if you don't duck before they take a picture, and in later stages love to hang out with other Earthlings who will definitely get in some free hits. Having Tourists around also means that the "Jump and carpet bomb the ground with jars" strategy that works with a lot of enemies is the worst idea because that's when you're most vulnerable to a camera flash.
Any flying enemy count as well, including the fairy who moves rapidly and drops bags of some kind of pixie dust or something that makes you laugh uncontrollably (having the same effect as the tourists' camera flash) the duck on a magic carpet who moves even faster and dive bombs you, and the ghost cows that are basically flying, teleporting boogiemen. The general strategy for any of them is to just constantly toss your jars into the air hoping they'll fly into them.
The harps when you get an extra life in the original game.
Panic on Funkotron has a fanfare whenever you pick up a jarred Earthling.
Period Piece: There's no way a game like this, with all it's bright and colorful, family-friendly, Totally Radical hip-hop aesthetics could've been made outside of the early 90's (which is probably part of the reason why Mission to Earth flopped so hard.)
Scrappy Mechanic: The Goddamn RANDOMIZER! It randomizes every present (including itself) so you don't know what's inside any of them and have to start again from scratch.
As a result, identifying the randomizer is the Crowning Moment Of Awesome of any run of the first game. Sure, to the untrained eye, all you do was talked to a man dressed as a carrot who caused the word "????????" to turn into the word "randomizer", but still.
There are actually circumstances in which the Randomizer can be an asset — since it changes not only the appearance of presents but their content, if you have an inventory full of undesirable items, spinning the roulette wheel is bound to turn some of them into helpful things instead. Of course, this is better used near the end of the game where you won't have to worry about re-identifying many presents.
Very rarely (that is, less than once per game), the characters will sneeze, which can spoil an attempt to sneak past sleeping enemies or sneak up on Santa. If they've recently drunk a Root Beer, they will continuously burp for a while, which also renders stealth impossible.
You can use a Game Genie to drink dozens of root beers all at once. The effect is cumulative; you burp all game.
Earl's pants may suddenly fall down, forcing him to spend a moment pulling them back up.