No. Capital losses must be claimed each year and can reduce ordinary income by $3000. The remainder (if any) is carried to the next year. Failing to claim the loss in any year prohibits the remainder to be carried forward and it is lost.
That is correct. It does not mean that the carryover is always reduced by $3,000 if you did not have that much taxable income to reduce, but it still must be reported each year. The carry-over amount can only come from the carry forward amount of the prior years schedule D. If no prior years return was filed with a schedule D then there is a zero amount to carry forward.