1) The best way to reduce heat, assuming you insist on a linear regulator, is to use separate secondary windings to produce the raw voltages on your regulators. 20 VAC is OK for 25 volts, but 12 - 15 VAC for the 12 volt supply and 6-8 VAC for the 5 volt supply is much better. Of course, then you need separate rectifiers and raw DC capacitors, too.
2) If you must use the existing transformer, zeners are (technically) an excellent choice. The 1N3998A is available from Digikey, and you can use one for the 12 volt supply and 2 in series for the 5 volt supply. As long as output current is less than 1.5 amps you'll be fine. Of course, you'll need heat sinks for them, since each will dissipate 6 watts per amp. Of course, you may not be happy with the $25 price tag. That's per zener.
Power resistors are certainly possible, and about 10 ohms and 20 ohms will work. (They cannot be in series as the diodes are in your schematic). They'll need to be at least 10 watts and 20 watts for the two values I've given, and more is better. They'll also need cooling.
The problem with any linear regulator is that raw DC current must equal output current, and that current times the voltage drop from raw to final voltage is the amount of heat that must be dissipated in the supply.
3) In the charge circuit, it's true that the resistor will drop some voltage, but in direct proportion to current being drawn. So, as the battery charges, its voltage will rise, and the charge current will drop. At the very end of the charge, the battery voltage will essentially equal the charge voltage, the current will be very small, so the resistor drop will also be small. In fact, it is the difference in voltages divided by the resistance which determines what the current will be.
Most hot-knife tools use a fairly beefy metal cutting tip. This is often a strip of wide nichrome ribbon - wide for mechanical strength and nichrome to make it easier to heat. You can use materials other than nichrome but these materials usually require significantly more current.
Because the cutting tip / cutting head is both small and beefy, its resistance is quite low. You therefore need a fairly-low voltage but lots of current.
To build your won power supply / controller, you need to start at the cutting tip. Determine what size of cutting tip you need. Then either figure out or measure how much current it's going to take to get it hot enough to cut your material. When you have an estimate of both the resistance of the cutting tip and the current required, you can choose a suitable operating voltage.
Most hot-knife cutting tools that I've seen and worked with use anywhere from 5 to 25 Amps at voltages ranging from about 1 to 3 Volts.
Note that I am talking about a hot-knife cutter here. There is a similar class of tools that use a long wire instead of a short ribbon for cutting. These are used for cutting foam board and sheets. Because the cutting element isn't small and beefy, these usually require less current but significantly higher voltage. But the principles are the same.
When you have determined what voltage and current you need, either purchase or build the power transformer that you need. There is a plethora of DIY articles that will show you how to re-purpose the power transformer from an old microwave oven for this and similar uses.
Finally, you need some form of current control to set the temperature of the cutting tip. Because the total power involved is relatively small (500 Watts or less), a triac-based dimmer works well. Again, you can either build or purchase your own control. However, I find that the speed controls used for ceiling fans work well in this application. They are usually quite inexpensive and because they are designed for inductive loads, have the appropriate snubber circuit that allows them to work with your transformer.
The dimmer controls that I've purchased from eBay have worked extremely well and they cost me significantly-less than what I could build them for. In fact, the price for the completely assembled and working units that I purchased recently was less than what I could have purchased the individual parts for.
Best Answer
It's quite far from certain death (indeed, it's voltages "generally recognized as unlikely to be hazardous", but we just had a big long discussion of that starting from a more hazardous and less useful premise.) Summary - you're a lousy conductor, you won't carry several amps at those voltages. Doesn't mean you can't make a big messy spark, or start a fire, but it won't electrocute you if you leave the box closed, unless you walk off into very marginally possible or probable scenarios.
The simple answer if you want a small-amperage supply is some in-line fuse holders with fast-blow fuses of whatever size floats your boat. If a quarter-amp suits your needs, it does not matter if the supply can provide 50 amps. You won't pull much more than 1/2 amp for a very short period before the 1/4 amp fuse vaporizes.
The short-term probably more expensive, possibly long-term cheaper option is DC circuit breakers.