Please see here about the motherboard: http://tinyurl.com/ooos8qu .
There is no urgent need to change the motherboard if you can now play your USB mic recordings at correct pitch. Changing the motherboard may not be easy on a notebook. You should ask Packard Bell.
Motherboard sound is never going to be as good as a new PCI sound card or USB sound card. You probably made a bad choice getting a notebook that you probably won’t be able to add a PCI sound card to and a USB mic without headphones that won’t be usable in either PCI or USB sound card. Because of those choices you will not be able to use the same device for playback and recording.
Perhaps you should do some overdub recordings - record or import a backing then record your singing against that. Do that and play to see if the beat in the tracks drifts apart over the length of a song. If the beat doesn’t drift then it may not matter that you are using different devices for playback and recording.
Remember to record with Windows Direct Sound, Exclusive Mode. 44100 Hz or 48000 Hz Project Rate should not matter. If the equipment is correct you should not be able to hear any difference between them. Record one bit of singing at 44100 Hz in one project and another bit at 48000 Hz in another project. Play one project then the other. Can you hear the difference?
Here is an interesting topic for you to read http://tinyurl.com/7byguts .
I deleted your repeat questions about sample rates and bit depths. They were just copy and paste from your first topic. Please read your first topic for the answers to those questions https://forum.audacityteam.org/t/pitch-of-voice-problem-solved/31599/1 .
Gale