Lovely to see someone rescuing reptiles! Theyāre often quite overlooked, especially with how finicky their care needs can be- and their long lifespans, as your little critterās previous owners dealt with.
Leopard geckos arenāt one of the species Iāve a special interest in, but I do know a bit due to spending time in adjacent exotic communities and there being quite the overlap in owners. The weird onion/turnip-y tail is totally normal: they more or less always grow back a bit strange after having been dropped in the past, though at least they grow them back, unlike cresteds! (Those little guys just sort of have mouse butts afterwards, and it can cause some balancing issues, but they tend to cope just fine.)
He does look a bit chunky- but itās a bit hard to tell from the angle of your photos. Basically, there should be a smooth taper to the body- itās when they sort of look like the pear/tear drop shape of milk drunk kittens when viewed from above, then you know theyāre too chunky. A kitchen scale can be useful for weighing them, especially if youāre going to be putting him on a diet.
They donāt strictly require UVB lighting, and it can be an issue for albino or more light sensitive morphs: but it can be helpful in maintaining good health. One of the worst afflictions from improper care reptiles can have is something called Metabolic Bone Disease (MBD), which is usually caused by improperly balanced calcium, phosphorus, and vitamin D3 due to poor diet and/or lighting. Basically, it results in permanent bone deformations- they get all squishy and malformed, and itās a very painful and debilitating condition that is unfortunately common in the hobby.
Itās for this reason itās recommended that calcium be dusted onto all meals, a once a week multivitamin provided, and D3 be supplemented distinct from calcium either on a twice-a-week basis (if no UVB light is used), or a few times (1-2 times) a month if a light is used. Your calcium supplement should not be bundled with phosphorus- their diet of bugs should provide plenty of that. Arcadia and Repashy are both reputable brands that offer easy dusting solutions.
Theyāre a bit trickier than crested geckos, for whom a complete diet is made in whatās more or less fortified fruit smushy paste like baby puree packets. Live feeder insects should be gutloaded for best practice- basically, feeding them yummy vegetables or fruits and bits of scraps like that before being culled for feeding.
Dubia roaches and crickets are a common choice for staple feeder insects- (crickets being so chitinous can sometimes cause concerns with impaction as a sole food source, especially in younger babies, hornworms and earthworms are common ātreatsā and can be nicely hydrating rotations as staples, though some like superworms or waxworms are not suitable as a main staple due to nutritional imbalances. You can keep a box of dubias in a moving tote with some egg carton packaging for them to hide in, and theyāre super easy to rear and feed. Crickets tend to be noisy and stinky, and also run away super fast, annoyingly.
Leopards do tend to be calmer by nature, and less likely to drop their tails than cresteds: but usually seem happiest if theyāre given the choice to hang out on your hand/with you, rather than being scooped up and plucked up. They should not be flipped over (canāt breathe in that position easily), and they are a bit bitey sometimes, something thatās hilariously common with little hunting exotics: but if that happens, a little droplet or spritz of water onto their heads tends to get them to release, rather than yanking or shaking them off (which could hurt them.)
Try not to grab them by the tail or around the head, since that tends to startle them pretty bad- which can lead to biting or tail dropping. If the tail does drop again- the wound should kept clean and dry, but will generally close over just fine on its own. Dilute betadine (in about a 1:10 part ratio of betadine to water) can be used to irrigate/flush the wound to prevent infection, and swapping them over to a papertowel substrate will help prevent loose soil from getting in there. An antibiotic ointment like Neosporin specifically without additive painkillers/analgesic ingredients, as those can be toxic to reptiles can be used over the stump while itās all gnarly. Hopefully this doesnāt happen, though! But just in case it does.
They live in fairly arid conditions, and donāt really tend to have as many shedding issues- but a shallow soak (tupperware container lined with a bit of paper towel to give them the safety of ātractionā to keep them calm, and a little bit of lukewarm water, about a third of your finger high and under monitoring) or a very careful, gentle roll of a dampened cotton bud can help loosen stuck shed retained about the eyes or in between toes, or the tips of tails. The shed should come off easily, and should never be forced off.
Forgive the blathering post- this is just information Iāve off the top of my head when it comes to these little guys. I hope you two have a beautiful life together, congratulations on the new pal, and thank you for rescuing a reptile in need. Purchasing from ethical breeders is how weāre able to sustain healthy captive bred populations, but there are always already existing little guys in need of a good home.