Doing the Tawaf circuits consecutively is a condition of it being valid, according to the most correct scholarly view. However, a brief interruption between them does not matter. If the Iqamah for prayer is given when you are doing Tawaf, then you should pray, then complete Tawaf from where you left off in order to pray.

It is permissible for the pilgrim to take a bath after wearing Ihram for the sake of cleanliness. It is also permissible to change his Ihram garments for cleaner or newer ones.

If the sun set and the adhan was given for Maghrib on the thirteenth of Dhul-Hijjah before you stoned the middle jamrah or the largest one, then the time for stoning the jamarat had ended, and whatever stoning was done after the adhan is invalid, so you have to offer a compensatory sacrifice. The same is also required of your father. So you can delegate someone to slaughter two sheep on your behalf in Makkah, and distribute the meat among the poor of Makkah.