For starters IMHO it is always better to use a release (e.g. 'TinyCore-4.4.iso') instead of a release candidate for such an exploration. The first obvious things to ensure are that the download was successful (by comparing the MD5 checksum, e.g. 'TinyCore-4.4.iso.md5.txt') and then again that the burned CD-ROM checks out (i.e. that the MD5 checksum matches as well).
With TC 4.4 a refreshed kernel build got introduced (i.e 3.0.21 instead of the previously used 3.0.3), which included a small adjustment for the use of older HW (i.e. enabling of 'CONFIG_PATA_LEGACY'). This should have rather increased the ability of TC to support older HW, and IIRC there were confirmations to that effect from some users. Nevertheless it would be an option to test a previous release (e.g. 'TinyCore-4.3.1.iso') for comparison.
Now to the question of capturing the actual boot failure: Does your system by any chance have a serial port and do you have access to another computer with a serial port and a null-modem cable? In this case you could simply use boot codes like 'debug console=ttyS0' to redirect all messages. A simpler option (of typically lower quality) is to increase the number of lines on the screen (e.g. via a boot code of 'vga=6', but you might want to use 'vga=ask', and pick one which shows more than 25 lines). You could then take a picture with a digital camera that you could either attach here or have it hosted somewhere and put e reference to it in your next post. Unfortunately reading these "squeezed" screen shots can be a bit tricky, so please make sure that the picture is sharp enough.