Praise be to Allah.
It is not correct to pray Taraaweeh and then ‘Isha’. You could have joined the congregation with the intention of praying ‘Isha’, then when the imam said the tasleem after the two rak’ahs, you could stand up and complete the last two rak’ahs. You cannot pray qiyaam before ‘Isha’, rather it should be done afterwards, and after the regular Sunnah of ‘Isha’. The prayers that you did were simply voluntary prayers and not qiyaam.
Shaykh ‘Abd al-‘Azeez ibn Baaz was asked:
If a Muslim comes to the mosque and finds the congregation praying Taraaweeh and he has not yet prayed ‘Isha’, should he pray with them, with the intention of praying ‘Isha’?
There is nothing wrong with his praying with them with the intention of praying ‘Isha’, according to the more correct of the two scholarly views. When the imam says the tasleem he should stand up and complete his prayer. It was proven in al-Saheehayn from Mu’aadh ibn Jabal that he used to pray ‘Isha’, then he would go back to his people and lead them in ‘Isha’ prayer, and the Prophet (peace and blessings of Allaah be upon him) did not denounce that. This indicates that it is permissible to offer a fard prayer behind someone who is praying a naafil prayer. In al-Saheeh it is also narrated that in the fear prayer, the Prophet (peace and blessings of Allaah be upon him) led one group in praying two rak’ahs then he said the tasleem and led the other group in praying two rak’ahs, then he said the tasleem, so the first one was his fard (obligatory) prayer and the second was naafil for him whilst they were praying fard. And Allaah is the source of strength.
Majmoo’ Fataawa al-Shaykh Ibn Baaz, 12/181
And the Shaykh also said:
The Sunnah is to pray tahajjud – in Ramadaan and at other times – after the regular Sunnah of ‘Isha’, as the Prophet (peace and blessings of Allaah be upon him) used to do that. It makes no difference if the tahajjud is done in the mosque or at home.
Majmoo’ Fataawa al-Shaykh Ibn Baaz, 11/368
With regard to the Taraaweeh prayers that you have missed, you have the choice. If you want you can do them and if you want you can leave them. Taraaweeh is a naafil prayer, so you do not have to make it up as is the case with the five daily prayers.
And Allaah knows best.