1) I thought I read this somewhere, just want to confirm

If a soldier has elite status, and you lose on the alien die roll, that soldier just loses his elite status and does not die, correct?

2) The chief scientist has an asset (Can't remember name off top of my head) that let's you spend a salvage to roll an extra die on a task.

Do you get that extra die until you get it, or lose, or is it just a one time use?

1) Incorrect - if an elite soldier is assigned to a task where you roll a loss, then the soldier dies (as normal) and also loses elite status (allowing that elite marker to be reused for another soldier later)

2) All extra dice work for all rolls against a given task - the Laboratory is no exception. If you check the wording on the card, it says something like "add 1 die to each roll during a research task" - "each roll" means each roll, not just one. You can't add it retroactively - rolls you've already made have already been made - but you keep the extra die until you complete the tech or stop rolling for it for the turn (usually because you rolled a loss and exhausted all the assigned scientists)