First, after recording a part, and if I want to record a short part OVER TOP (to fix a few notes), once I press 'record,' the part I already recorded beforehand goes silent.  How do I fix that?
 
Second, when I split a track and delete the part I don't want, it also deletes a previously recorded part under it... it does not delete ONLY the clip on top.  How do I delete only the part I want while not affecting a clip underneath?

Do you mean the lead in is silent or the replaced part auto muted?
 
Sounds like you need to check your recording mode and settings but until we know exactly what you mean it's hard to say.

Yes... the part I already recorded goes mute.  I still can hear the NEW part and what I'm playing, but it mutes the part already recorded.

I should add something... if I record a part over another part, the old part also goes mute on playback, but the new part can be heard.
 

OK... I found the problem.  I used to run guitar tracks pro, which did NOT have different record options.  Once I selected 'sound on sound,' both my recording and 'deleting' problems disappeared.  Thank you!
 

No worries, not that I really done anything. It's just that there are several options for what happens to recorded material and whether you hear it or not, hence my questions about your question.