Just to answer your questions roughly under the assumption that you will pray as fadd (alone):
Is it permissible to read in prayer just part of a Surah which I have memorized from the Quran so I can use it as a repetition?
Yes it is permissible, however you should be aware that reciting with the Qur'an in hand is hardly allowed (at least frowned upon) for fard (ordered) prayers, but permissible for sunnah/nafl (optional) prayers.
If it is so, what to do if I forget how to continue?
If you forgot the continuation you could move to ruku' or if you haven't recited much recite from some other surah you memorize. You could also keep the moshaf open in front of you and recite from it.
Should I pick up the Quran, find the part and continue reciting while reading from the Quran in paryer?
You shouldn't do much movements which are not related to the prayer, like searching in a book etc. as it is distress and interruption of your worship. However the different maddhabs have different opinions on what is allowed or not, like opening a door or going back or forward etc.
I found this Question. One problem I have with this answer is that the Imam in Taraweeh reads the whole Quran for the month of Ramadan. So he continues to read from the Quran from the last ayah where he stopped in the last rak'ah or prayer.
First there's no clear ruling on how to pray taraweeh it is rather a custom that we actually complete the quran (Is it necessary to complete the Quran in taraweeh from one Masjid?), basically we could recite what ever comes to mind. And is not even necessary to pray taraweeh in a mosque (Is it obligatory to pray Taraweeh in the mosque?). So basically the Imam could after standing up from the first rak'a recite from any other starting verse in the Qur'an.
You may find further details in posts like:
Surah recitation during salah
Starting the recitation from middle of the ayat
