I need to build a high current, low voltage buck converter

If you have a fundamental understanding now, you will be an expert when you finish this design!

Are there any problems with the combination of components? Well, there are things to watch out for:

AOT2140L: Like all muscular MOSFETS, the AOT2140L handles high current at a price: large capacitances. In this case, the gate capacitance is 10 nF, so you will need as much current (and voltage) as you can get in the drive side, and you have to have an adequate bootstrap capacitor and diode. I might add a separate external Schottky. You'll need short wide traces not only for your current paths but also for your gate drive power/decoupling traces and the gate drive signal itself. Any inductance or resistance in the gate path will cause the part to oscillate at the switch point, costing efficiency and heat.

TPS28225D: You won't be able to run this circuit with 4.2 volts, because the internal bootstrap diode in these parts require 1 volt, leaving only 3.2 volts for the bootstrap. Once again, an external Schottky would help some, but even so this will be too close to the FET gate threshold voltage and the switching times will suffer. You can try running the device with a separate, 8 volt power source. Use big decoupling capacitors near the part, because you have to supply 2 amps to the gate.

MPXV1D1770LR68: Be sure and calculate the maximum current you expect to see in the inductor; you can't use the average. If your instantaneous current approaches the saturation point, you will experience high current spikes. This same warning applies to your battery/power capacitor combination. Also, there will be a considerable resistance increase in the inductor winding due to skin effect; do not expect the apparent resistance to be that in the spec sheet.

Should I design the test circuit on a PCB or can I lay it out on "proto-board"? No luck here.

Your best chance is to use a circuit board that you lay out yourself, with wide areas for traces and possibly conductors on multiple traces connected by multiple vias. Remember, you are switching 30 amps, so both resistance and inductance of your current paths come into play. You'll want low ESR decoupling capacitors, and I might lay out for a snubber just in case.

Is there a different design I should consider? (Multi-phase, serial coupling, other.) Of course there is more than one way to skin this cat, but I see no problem with your approach, and you should have some fun with it.

Here is a simulation of a simple buck converter that will provide about 30 A into a 40 mOhm resistive load. The MOSFET I chose dissipates about 15 watts at 50 kHz. This is mostly conduction loss, as this device has 25 mOhms ON resistance. Other devices should be available to improve this, or you might add one or two in parallel.

Using an IRF7210 brings its dissipation down to about 9.5 watts. But there is also about 14 watts in D1. So for 46 watts into the load, efficiency is only about 50%. A 25A Schottky MBRB2545CT brings its dissipation down to about 9.5 watts.

The IRF7210 is obsolete and cannot handle 30A, so an FDMC610P should do the job nicely, and is in stock for a couple bucks.

A better approach would probably be a push-pull driver into a center-tapped toroid with about 20 turns, and 2-3 turns on the secondary as needed for the low resistance load.