If a fruit that has grown on a tree during the shmita year and is only picked in the 8th year (not a shmita year) does it have a kadosh (holy) status since it grew and was ready to be eaten during shmita?
Rambam Shmittah 4:13 (translation from Chabad.org)
וכן פירות שביעית שיצאו למוצאי שביעית בתבואה וקטניות ואילנות הולכין אחר עונת המעשרות. והפרגין והשומשמין והאורז והדוחן ופול המצרי שזרעו לזרע אחר גמר הפרי. והירק אחר לקיטתו:
Similarly, when the produce of the Sabbatical year is reaped in the eighth year: With regard to grain, legumes, and the fruit of the trees, [the ruling depends on when the produce reached] the stage when tithes are required to be separated. When rice, millet, poppy seeds, sesame seeds, and Egyptian beans are sown to produce seed, [the ruling depends on] when the produce completes its growth. [The ruling regarding] vegetables [depends] on the time they were harvested.
So it depends what kind of plant it is and (in some cases) how fully grown it was when the new year passed.
(This answer is not intended to address Etrugim.)