What do I need in order to build and test a ferrite rod antenna for a specific frequency?
You asked a bunch of questions that are really too broad taken together, so I'll just answer what seems to be the underlying question about how to make a tuned ferrite rod antenna.
Basically a ferrite rod antenna is a resonant L-C circuit. The ferrite rod and the coil wrapped around it form the inductor, and you connect a deliberate capacitor accross it. The Q can be fairly high since it is limited only by the resistance in the inductor coil and any losses in the ferrite. Make sure to get ferrite rated to a frequency well beyond the one you want it to resonate at. At 457 kHz that won't be a problem.
The resonant frequency of a L-C circuit is:
F = 1 / 2π sqrt(LC)
When L is in henries and C in farads, then F will be in hertz. Of course you can rearrange this to get any of F, L, or C from the other two. For example, to find the inductance to resonate at 457 kHz with a 10 nF capacitor, you need
L = 1 / (2πF)² C = 12.1 µH
Since your frequency is fixed, by solving for just one L-C pair, you can easily get others. For example, if you wanted 10x the inductance, you'd have to use 1/10 of the capacitance, or 1 nF and 121 µH.
The best way to get the right inductance is by experimentation. Yes you could in theory get the data for the ferrite rod and do a bunch of calculations to determine the number of turns, but it will be easier to simply try something, see where you're at, and adjust iteratively until you get the desired resonant frequency. From the numbers above, a capacitor in the 1-10 nF range should work well, as 12-120 µH is doable. I'd probably aim for something in the 50-100 µH range. Do the math, get a suitable capacitor, and start winding. Capacitors aren't usually that accurate, so start with the final cap and adjust the inductor until you get the desired resonant frequency with that cap.
I don't know how big your ferrite rod is, but as a wild guess, start with around 50 turns of magnet wire and see where you're at. Something like 28 gauge enamel coated wire will probably be about right.
There are various ways to find the resonant frequency. I'd probably start with a function generator, resistor, and scope. Feed the L-C tank circuit (your inductor with the cap accross it in parallel) from the function generator thru a resistor, and look at the voltage accross the L-C on the scope. There will be a sharp amplitude peak at the resonant frequency, and it will be nearly 0 elsewhere. Sweep the frequency by adjusting the function generator dial to find the peak, then see what the frequency is. I would have the scope tell me the frequency instead of trusting the function generator dial. Those are notoriously inaccurate, unless you have a precision calibrated frequency generator.
If the resonant frequency is too high, add more turns. If too low, take a few off. Iterate until you get it just right. Once you do, put some hot glue or epoxy on the windings to keep them from moving around.
Now you have a sensitive magnetic antenna tuned to the frequency of interest. The rest is a amplifier followed by a detector, but that's too much to get into for this question.
I've built up a few long wave receivers for airport beacons here in Australia. They usually operate between 200 and 450 Khz, so they might be within the ball park of what you are looking for in terms of the L/C component of your receiver. Most simple receivers of this type are based on the MK484 or TA 7642 IC's, used in order to replace the ZN414 IC.
As for the tuned circuit, athe better the quality of components used, the better your results will be. AM and below are generally pretty noist bands in terms of hum, hash and other unresolvable signals. If you can, obtain a tuner cap that uses ceramic insulators rather than paxilon or other not so good materials. Ceramic is to tuner caps what glass is to high voltage cables - it prevents signals leaking to earth, and signal loss is something that you don't need.
So, a good tuner cap of say 500 to 600 pF per gang, and if it is 2 or 3 gang, even better - you will use less wire. With a lerge value cap and a good ferrite rod (the longer and thicker the better) try winding around 120 turns onto the rod. Use litz wire if you can get it.
This can be scavenged from defunct antenna coils or old disused IF transformers - more than enough for a 120 turn coil can be retrieved froma 2 bobbin IF metal can device, but you'll have to be careful about loosening the outer layers, as they are often held intact by a brownish coloured glue.
This glue can wreck the first two or so layers if you're too rough in handling it, so use a scalpel or a hobby knife to scrape the dried old glue away - gently, and retrieve as much wire as you need.
Good ferrite rods can also be rescued from old AM receiversw - just remove the rod and its supports, and remove the existing coil and replace it with the one you need to wind. Alternatively, use the coil that is on the rod, and simply use two or three tuning capacitor gangs wired together, to get the capacitance low enough for the 457Khz f0 that you need to receive at.
Avoid cheap polyvaricon (plastic) tuner caps and cheap, short ferrite rods, as you end result may be sdisappointing if you decide to use them.
Pursuing an advanced project, designing a receiver from scratch, A) without knowledge, B) without equipment, C) without prior experience? Be careful, since major failure at first projects can steer students away from electronics entirely.
Go and build lots of beginner projects first. To succeed at the avalanche receiver, you should already have built simple transmitters/receivers as kits or online projects with schematics, and already know the answers to your questions.
If you really insist on tackling advanced projects, instead do circuit-mods: alter a commercial AM radio, or find an online project/schematic which comes close to your needs (such as VLF radio receivers for hearing "whistlers" etc.