Is it valid to perform a two-rak'ah prayer between the Adhan and Iqamah at home, and a two-rak'ah prayer for ablution before the obligatory prayer?

It is permissible for the worshipper to perform voluntary prayers between the Adhan and the Iqamah, due to the saying of the Prophet, peace and blessings be upon him: "Between every two adhans (Adhan and Iqamah) there is a prayer." Your two rak'ahs of prayer after the Adhan are correct, but one should be diligent in observing the regular Sunnah prayers: four rak'ahs before Dhuhr and two after it, two rak'ahs after Maghrib, two rak'ahs after Isha, and two rak'ahs before Fajr. Your two rak'ahs of prayer after wudu (Sunnah of Wudu) are correct, and this can be achieved by performing the regular Sunnah prayer.