Is it permissible to perform ghusl while covering the front and back private parts to avoid invalidating wudu, and does touching the private parts invalidate ghusl, and are wiping and washing the back private part obligatory during it?

What you do—washing the private parts, then performing ablution (Wudu'), then washing the entire body (Ghusl)—is correct, while being careful to start with the head, then pouring water over the right side, then the left. Touching the penis or the anus during Ghusl does not invalidate it, but it does nullify Wudu' if it occurs after washing its limbs, so it must be repeated. Ghusl from major impurity (Hadath al-Akbar) suffices for Wudu' as long as no nullifier occurs. Rubbing (Dalk) in Wudu' and Ghusl is recommended, and washing the anus during Ghusl is obligatory.