Were the lords of Andunie also from House of Elros, or was Elendil just carrying on the kingship of Numenor to Gondor? Was Elendil a blood-related heir of Elros, or just a new line formed after the downfall of Numenor?
Elendil is a direct descendant of Elros, through Elros' great-great granddaughter Silmariën. This is described in Appendix A of Return of the King:
These are the names of the Kings and Queens of Númenor: Elros Tar-Minyatur, Vardamir, Tar-Amandil, Tar-Elendil
In the days of Tar-Elendil the first ships of Númenóreans came to Middle-earth. His elder child was a daughter, Silmariën. Her son was Valandil, first of the Lords of the Andúnië in the west of the land, renowned for their friendship with the Eldar. From him were descended Amandil, the last lord, and his son Elendil the Tall.
Return of the King Appendix A "Annals of the Kings and Rulers" (i) "Númenor"
The Thain's Book has transcribed the full family tree:
Yes Elendil was of the Line of Elros in the female line. His cousin was the last king of Numenor. By the time of Elendil there were perhaps many people related to Elros. It is obvious in the genetic makeup of the line of Andunie,
To Elros, who chose to be a king of Men, still a great span of years was allotted... all his line, the kings and lords of the royal houses, had long life even according to the measure of the Numenoreans."
Elendil was killed by Sauron when he was 322 and this was during the waning of the lifespan of the Numenoreans. This waning was much faster in the rebels which included the King's house since his cousin Ar-Pharazon felt his end coming when he was only 200 years of age.
Otherwise there is also this piece of text:
Highest in honour after the house of the kings were the Lords of Andunie; for they were of the line of Elros, being descended from Silmarien, daughter of Tar-Elendil, the 4th king of Numenor.
In any case, compared with other Numenoreans the line of Elendil had a clear distinct advantage "in life-span, vigour, or ability" [Note 27; Aldarion and Erendis].
Elendil was the high king of Arnor and of Gondor and the high kings took up their seats in Arnor giving the rule of Gondor to their children, or nephews. The elder line stayed in the North and the younger line stayed in the South which usurped the rightful rule from the elder line.