Good 'ANS'WER from 'ans' (and answered two out of three questions).
1) What was the original name of Trishanku ?
Trishanku, the son of Prithu was a king in the Solar Dynasty, the dynasty of the great Hindu God Rama. Trishanku's original name was SATYAVRAT and this name Trishanku was his pseudonym.
"tasya satyaratha nama bharya kaikeya vamshaja |
kumaram janayamasa harishchandram akalmasham ||" 1-13-24
Satyavrata's wife is lady Satyaratha, a descendant of kekaya lineage, and she begot the impeccable king Harishchandra.
2) Why Trishanku called as Trishanku ?
He got this because three sins were attached to him tri shankas as said in Harivamsha.
"pituh cha aparitoshena guroh dogdhrI vadhena cha |
aprokshita upayogat cha tri-vidhah te vyatikramah || 1-13-18
"You transgressed in three ways... you hurt the feelings of your father in two ways - by using an unsanctified food earmarked for manes, and by abducting an bride; now, by killing a sacred cow of your own family priest..." vashistha cursed satyavrata in this way.
One by his father's damnation - when he misbehaved in his kingdom, his father being a righteous king exiles him to a wretched place. Satyavrata earlier kidnapped the daughter of a brahmaṇa when she was being married, his father cursed him to become a candala, (Srimad Bhagavatam 9.7. 5 &6). (When his father was exiling him, Trishanku took umbrage at Vashishta, because Vashishta did not come to his rescue).
The second blemish is due to the killing Vashishta's milch cow, knowingly. (Trishanku kills that cow, when he did not get any food for the day to feed the son of Vishvamitra. At that time, he had to protect the wife and three sons of Vishvamitra, as Vishvamitra has gone away to a distant place for his ascesis, leaving his kingdom and family. Then Trishanku comes to the rescue of his family. With that gratitude, now Vishvamitra tries to requite this king in sending him to heaven).
The third damnation is for his usage of unsanctified meat of his kill (cow meat, Harivamsa 1.13.5). Thus, Trishanku is the one who carries 'three social blemishes' on his head, and added to these is the curse of the sons of Vashishta, in making him as a profaner.
3) Who labelled him first ?
"evam trIni asya shankuni tani drishtva mahatapah |
trishankuh iti ha uvacha trishankuh iti sa smritah" || 1-13-19
'Because satyavrata committed three offences, VASHISTHA labelled him as tri-shanku as such he is generally identified with the name trishanku.'