Ballpark 60 days. Any longer its efficacy starts to get questionable. There might be new developments in this area, but 60 days used to be the rule of thumb.
How is it stored? Temperature dependent? Special containers, ie vials or prefilled syringe? How is it administered? IV of IM?
Normally in powered form for storage for some time (60 days). Antivenin is normally freeze-dried and stored. Whenever it is needed a diluent is added. I do know that for short trips to the bush some polyvalent antivenins do come in small vials that have already had the diluent (bacteriostatic water) added, but if not used those ones have to be thrown away on getting home.
Can it be self-administered?
You'd have to be a trained medical professional to do that (or a the very least trained in how to use it). In most nations a person cannot just have some laying around, and one cannot just go and buy antivenin(again in most countries). The risk for allergic shocks leading to anaphylactic shock and death are too great. And what you use for a coral snake is different from what you use for a Cottonmouth. Also many people might inject themselves even after been tagged by a non-venomous snake. Technically speaking a competent person can do it, but the possibility of that happening is not too great and the person better know what they are doing. Too many people would kill themselves.
And one more reason why self-administering antivenin is a death wish. For major bites over 30 vials may have to be used. Over 30. I know that for Mamba bites double the dosage for cobras is used. Those are going to be many vials. And guess what. There has to be a time interval between vials .....15 to 20 minutes per vial! Very few people would have the presence of mind to do that after being tagged .....and even if they did they might be unconscious or writhing in pain long before that. Basically when bitten a person needs to go to the hospital.
Your scenario with all of those kids being bitten. Can they really do that?
No. A single mamba would not tag so many people. It's venom is toxic enough, but the snake would have to possess amazing intelligence to know exactly how much venom to inject, and to measure it out. Obviously that is not possible. A single mamba bite to one person may contain enough venom to kill several people, and on the other hand it might be a 'dry bite' where the snake does not inject any venom at all. I know of 2 documented cases (that's important because there are all sorts of silly tales out there) of mambas killing several people, but 25 (and even much less than that)is not possible in the wild. Now, if you gave me mamba venom, a micropipette, and a syringe i could do it .....but don't expect a mamba to do it. For example against mice a single bite has enough venom to kill over a hundred ...when the mice are injected by scientists using carefully measured samples. However in the wild a mamba would inject a single rodent with enough venom to kill it several times over. To answer your question about the mamba killing 25 kids .....never happen in the wild. However give me one mamba, a jar with muslin covering to collect venom, a micro-pipette, and a syringe .....and after collecting one venom batch i can kill 25 kids easy. But that will not happen in the wild. The most i know (that is documented) is 9 herdsmen, a dog and 3 cows being killed by a single mamba, which is pretty impressive, but still not anywhere near 25. In most cases after a bite the snake will usually escape.
I transported a patient to the hospital with several bites from some sort of hemotoxic snake. (don't remember what kind of snake and I wasn't about to let that live snake ride to the hospital with me). He was a professional snake handler. I would have expected someone like him to carry his own anti-venom, but he didn't.
Again in many nations people are restricted from owning antivenins due to the risk of anaphylactic shock. In the US they are regulated by the FDA. Secondly due to the short shelf-life many herpetologists prefer not to bother keeping, and continuously refreshing, antivenin. It can be a real drag after the first 60 days. Another reason is depending on the snake a bite can be pretty serious (no pun intended). The dude you transported was bitten by a hemotoxic snake. In the US probably a rattlesnake. That normally means serious tissue damage, and that needs to be treated. Left untreated the person could easily develop necrosis even if the venom did not kill him. I've never seen a rattlesnake bite, but i have come across the results of being bitten by a puff adder (which also has a powerful hemotoxin) and the results are not pretty.
Anyway, lucky for snake bite victims in Houston, the Houston Zoo is directly across the street from the Medical Center. The zoo had all the necessary expertise and anti-venoms.
And that is the best thing to do. Go to a hospital (if you are bitten by a local snake a local hospital should have someone who knows how to attend to bites from snakes native to the area). The doctors will ensure you are not allergic to the antivenin (a certain percentage of people can literally be killed by the antivenin). Also they can make sure you were bitten by a venomous snake (most people think all snakes are dangerous when most are not ....even the venomous ones are normally just midly so. There are relatively few truly dangerous snakes), and that even if venomous it wasn't a dry bite (up to 20% of bites can be 'dry'). And again depending on the venom a person can quickly become unable to stay conscious or in control of all their faculties. And all in all it is good sense to go to a hospital. The problem arises when a person has an exotic snake. For example, if some kid has a Banded Krait from India or an Australian Tiger-snake or a North African saw-scaled viper. Most hospitals can be able to easily take care of a rattlesnake bite, but you cannot expect them to tackle some of the more exotic snakes. And big zoos normally have some in stock just in case a worker gets tagged, but even then there is the chance they do not have the specific antivenin needed (in most cases a huge variety of snakes can be covered with one type of anti-venin. For example Polyvalent can take care of many types of bites that relate to a specific venom type. However certain snakes require antivenins specifically targeted towards them). The firefighter from Dayton Ohio who was killed by the Gabon viper died because there was no antivenin in the whole USA for him. And anyways to treat a Gabon bite would have required loads of vials. All they could do is try to stabilize him and ease the pain (and they probably tried a last ditch attempt with some rattlesnake specific antivenin). The guy must have died horribly. However i am certain if he had been bitten by a rattlesnake he would still be alive today.
It is much easier to get a dangerous exotic snake than to get the antivenin required for its bite. It is also much cheaper.
For example the person you transported to hospital. Had he been bitten by a Rhinkals (a type of spitting cobra) i doubt they could have done anything for him. Zoos have several species of cobra, but virtually all of them have neurotoxic venom. Not the Rhinkals. Like all dedicated spitting cobras there venom is not pure neurotoxic but an eclectic mix that has a lot of cytotoxic properties (that way venom in the eyes would ahve a greater effect). And when bitten by a Rhinkals (or similar spitting cobra) the effects are more along the lines of a severe Rhinocerous viper bite. Many big zoos can provide cobra antivenin, few can provide for the Rhinkals, or for the black spitting cobra, or the red spitting cobra. However these snakes are popular because they are exotic, dangerous, and in the case of the red-spitter very attractive. And many dumbos thinl they only spit ....stupid! They have a bad bite, and if you survive it you will never forget it. I've seen photos of severe necrosis, amazing skin sloughing, and a whole host of other unpelasantries. Still, the snake is prime property, yet if bitten by one chances are there will be no antivenin (at least in time).
Hope that helps.