As regards the use of cheap/expensive batteries.
No easy answer, batteries are designed for differant outputs over differant timescales, so one battery might be able to put out lots of power in a short time, and another might put out less power over a longer time, but the total energy output of both may be the same.
Dry cell batteries contain chemicals called depolarisors, which clear up oxides that are deposited on the electrodes during use, but if you use the battery too hard then those oxides will form a thicker insulating layer over the electrodes, reducing output and the depolarisors cannot cope, often it is not the fact that the electricity producing chemicals have been used up that limits the life of a dry cell, instead it is either using it in the wrong application or using up the depolarisor chemicals.
Generally it is best to use any dry cell in low current applications and leave a longish break between uses, my radio/alarm goes off every day and runs the radio for maybe half an hour and yet the battery lasts for months, the 24 hour break between uses allows the depolarisors to do their stuff.I use alkaline cells for this purpose.
Things like camera auto-winders and flashes present problems because they may be stored for many months and then called on for an intense short period of use, and then stored again for months.It is a good idea to use lithium cells for this.
Lithium cells are expensive but they have a very long shelf life and are sometimes used in c-mos memory power backups.
There are units that ‘recharge’ chemical dry cells, in fact what they do really is to assist depolarise them which increases their life by allowing them to use up their ‘fuel’ more fully.
Mixing differant types of cell or cells with differant ages is a bad idea.
Batteries have an internal resistance(due to the polarisation of the terminals - yes its that important) which increases as the cell ages.The internal resistance limts the current that can be produced, and if a short circuit should occur then this will prevent the cell overheating and possibly exploding or at least leaking out chemicals inside the appliance.
When you mix differant types of cell, some of them may be designed to withstand the higher current, but others may not and these are the ones that can overheat.
Mixing the ages of cells means that you may end up with a cell in position long after its useful life, and although the appliance may be operating from the good cells, the old one may be leaking unkown to the user.
You can get information about the power available from a dry cell(usually in trade catalogues or in, say, Radio Shack catalogue), it will be in the form of X amps for X time, or more rarely in joules, obviously the bigger the number the better, but…
There will also be information upon the maximum discharge current too, so even if a battery is rated at 1Amp/Hour the maximum discharge rate may only be 200milliamps so in theory it should last 5 hours at that rate, but of course using it at that level may mean that the depolarisors will not work as well, you may then get a much shorter life out of it.As a general rule you do not wnat to use a dry cell at more than 2/3rds its maximum current, and remember to give it a rest too.
Nowt at all wrong with lead-acids in the hands of those who know, and the safety precautions are not that onerous.
Put any equipment in an environment among the uninformed, or just plain stupid and accidents will occur, one has to take responsibility for oneself by making sure of the facts, seems to be a dissappearing philosophy in an increasingly litigious society but it has worked for me up to now.