OpenMPT can handle pretty much any sample rates and bit depths, and in general the higher the quality of the samples is, the better of course. However, as with any sampler (no matter if hardware or software), there are some limitations to sample playback that may cause samples to sound strange. As your description is a bit vague on what the actual problem is, I will assume in the following description that this is your problem, because it is what confuses most beginners about sample playback.
If you play a single sample across the whole note range, it will typically start sounding strange the further away you move from the root note at which it was recorded and is meant to be played back at. For example, if you play a middle-C on a real piano and then move up one octave, the higher C will have different characteristics in its timbre than the middle-C one. If you simply take a single piano sample and play it one octave up, it will keep the same timbre but play it at double the speed and twice the frequency, and as a result sound very tinny.
You can either embrace this type of sound distortion as the typical sound of samplers, or you can do what all other sampling software and hardware is doing as well, assuming that you have suitable sample material available: Use multiple samples per instrument, spread over the entire note range. OpenMPT's instrument editor (https://wiki.openmpt.org/Manual:_Instruments#Sample_Map) lets you assign samples freely over the entire note range of an instrument.
QuoteWhat is the ideal sample rate or bit depth for OpenMPT: ??
Are there any specific settings I should adjust to maintain the original sound quality: ??
As mentioned above, the higher-quality, the better, but going above the mix rate (typically 48 kHz) won't help. Lots of samples are available in 16 bit or 24 bit resolution (the latter will be reduced to 16 bit by OpenMPT upon import), often at 44 kHz or 48 kHz sample rate, which are both fine for OpenMPT, but due to its heritage from oldskool home computer trackers, it has no problem with lower quality samples (there is a FAQ entry (https://wiki.openmpt.org/Manual:_Frequently_Asked_Questions#Why_do_old_modules_and_chiptunes_sound_muffled_in_OpenMPT?) why such samples may sound strange with the default playback settings though)
QuoteAny tips for organizing or optimizing sample usage in larger projects: ??
Do you mean your sample library or the samples within a song?