1. I would say it depends on the breed of rabbit you are comparing to, from what I've heard they are similar to the smaller rabbits less friendly rabbits. I've only had larger breed rabbits in the past (french lop mixes) and they were way more laid back then a chinchilla. Also it depends on what you mean by "tame", I would say at least 90% hate being handled and always will, it comes down to individual personality. Getting a chin from the pet store it depends on how nice they were to it while there, most workers even if they care don't have the time to gently handle them so they are generally grabbed and moved which givens them a good reason to not trust humans. For chins it generally takes as long as the mistreatment has happened to get over it. So say the chin has been in the pet store for a month, expect it to take at least a month of nothing bad happening to them before they even start to consider building a bond with you.
Chinchillas you have to earn their trust, which takes time, but most can eventually (after months or years) with gentle handling learn to tolerate being handled for short periods, like to be taken out of the cage to have some playtime, as well as being petted on their terms. Their trust is hard to earn and easy to loose, for most you are a threat until proven otherwise, and if you ever have to do anything to them, like chase them to get them back in the cage, grab them, give them meds or other treatments, etc, all of that can cause the chin to lose trust in you since in their mind you proved you are a threat and can send you a few steps back or even all the way back to square one in bonding.
2. Unlikely, again you have to earn their trust, and their personalities vary as much as humans, some are more outgoing and brave others are more skittish and scared, and everything in between. It is possible to find someone selling a chin that is willing to sit in your lap and but most don't no matter how well they are treated or how much they are handled. It takes most at least a week to settle in to a new home, but up to a month or more is not uncommon depending on personality and how they have been treated. Also even the more friendly ones don't often stay on your lap or shoulder for long, maybe a few minutes before hopping off to go explore and play. Out of the 14 chins I've had I had one that would spend longer periods of time sitting with me and hanging out on my shoulder.
3. Again it really comes down to personality and how they are treated. By getting a young chinchilla right from a breeder that has cared for them since birth you eliminate the likelihood anything bad has happened to them to give them a reason to distrust chinchillas. Most of the kits I got from breeders tolerated being handled fairly quickly. If you get them from a pet store like petsmart or petco, you don't know where they got them, they may have came from a breeding mill situation where they were never cared for well and then shipped off to the pet store for more mishandling. However it's also possible you can get a older chin from someone selling them and they be friendly and trusting fairly quickly. For example my current chin and his brother I got when they were already 3 years old, and they tolerated me handling them from day one bringing them home, but they were also some of the most laid back chins I've had, kind of "go with the flow" personality.
4. Chins need to be kept in temps 70F or below, so really there is no replacement for an AC if you live anywhere that ever gets above 70F. No evaporative coolers do not work for chins (unless you live in the desert, then they work sometimes), in most cases they can't cool it down enough, most can only lower it a few degrees depending on humidity, and if it's 90 or above you would need it to lower the temp about 20+ degrees. They work by adding moisture to the air, so they don't work very well if it's humid, above around 20%, and/or if the temp is too low below about 80 (which is too hot for chins). Also adding moisture to the air if it increases the humidity above about 60% risks the chin getting damp which can lead to fungal and mold infections since their fur is thick it doesn't dry well, which is why they need to bathe in dust not water. You also don't want air blowing directly at the cage, it can cause a respiratory infection, also since chins can't sweat like humans do air blowing on them doesn't cool them like it does us. Ideally you want to keep the temp below 70F all the time (I keep mine in the 60s), that gives you wiggle room for the temp to safely go up a little just in case the AC dies and you need to go to the store to get a new one or the power goes out and you need to wait for it to come back on or get a backup generator running. Age, health, and how active they are all effect max temp they can survive, so although most are ok up to about 75F, there have been some that died in the lower 70s. Also damage done when they over heat is cumulative to some extent, meaning even though they survived hotter temperatures once doesn't mean they will the next time especially if it happens again soon after.