Given that Force-sensitivity comes with midi-chlorian count, is there a minimum age for determining whether or not someone is Force-sensitive?
Is it like puberty in humans or does the person need a calling from The Force?
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There does not appear to be a minimum age for detecting Force sensitivity. As far as Legends is concerned, in the Wookieepedia page for the Jedi's Acquisition Division, it is mentioned that newborns are blood-tested for midi-chlorians, so that suggests there is no minimum age for it, though the canon is murky as to whether or not this always happens.
It is also mentioned that the tests are not always accurate and that is why the names are stored in the Kyber crystal so that Jedi recruiters can later determine who is suitable for eligibility once they are older. As Skooba mentions, one of the jobs of Jedi recruiters is to find unrecognized Force talents, indicating that the tests were not used everywhere, and/or were not totally effective on their own.
In most cases the Jedi found the children early enough to take them on as younglings, whether they found them by accident, or through their testing system, or some other way (such as suspecting Force talent through heredity or species-specific traits), and almost always on worlds that either were Republic worlds or affiliated with the Republic. Since the Sith recruited very little due to the Rule of Two, their methods of finding their apprentices vary more widely.
There also does not seem to be a requirement for any calling from the Force. In Canon, Asajj Ventress used the Force before she was ever taken on as a Padawan, and in Legends Valkorion was using his powers as a six year old (and there are other examples of the Sith using their powers early, usually discovering them during a negative event). In TPM, it is speculated by Qui-gon Jinn that young Anakin was using his connection with the Force unconsciously to aid his reflexes in the pod-racing.
A few examples from Canon:
Ahsoka Tano came from Shili, a Republic planet, but was not initially discovered until Plo Koon found her at the age of 3
Anakin Skywalker grew up on Tatooine, a non-Republic planet (ruled by the Hutt clan) and was not found until he was 9
Mace Windu was taken at the age of 6 months due to interesting circumstances which are described more fully in Legends but generally accord with the Canon
Both Obi-wan Kenobi and Qui-gon Jinn were identified early on, hailing from Republic words, and taken in as Younglings, in Kenobi's case at the age of 6 months
A few examples from Legends:
Shaak Ti also came from Shili, but unlike Tano she was identified by the planet's doctors as Force sensitive
Aayla Secura was discovered by Quinlan Vos at the age of 2 on Ryloth, a Republic world
Ki-Adi Mundi came from a non-Republic planet (Cerea) and was found at the age of four
Sheev Palpatine was born into a wealthy and well known family on Naboo (a Republic world) and yet was not identified, nor even detected, until Darth Plagueis found him at around age 17, even though he attended elite schools and engaged in public affairs. Palpatine was aware of the Sith from an early age, though there is no indication that he was aware of his own Force talents until Plagueis arrived
Plo Koon came from a family that was strong in the Force so his connection with it would have been suspected from the beginning, and he was soon handed over to the Jedi for training
In Legends, Yoda was completely unaware of his force abilities until he was found by Jedi Master N'Kata Del Gormo, but he was not young at the time (he crashed landed in a ship that he was using to seek employment elsewhere). In Canon he is stated as having joined the Jedi Order as a youngling, though with scant details
Maul was identified as a child by Darth Sidious, but lived on Dathomir, a planet that was not part of the Republic
It seems unlikely that there is a specific age.
Before the Empire took power the Jedi search the galaxy for children to recruit. From Wookieepedia:
A specialized field of study undertaken by those Jedi Knights that desired to become a Jedi Sentinel, most Recruiters were retired Jedi investigators, turning their Force abilities of tracking and detection to locate unrecognized Force-sensitives. Working as members of the Acquisition Division, recruiters studied the data turned over from Republic med-centers and the Kyber memory crystal in order to track down potential recruits.
Jedi younglings, also known as Jedi initiates, were Force-sensitive children who underwent their training in the Force, typically at the Jedi Temple on Coruscant. Between the ages of four and eight, younglings were divided into ten "clans" of up to twenty children. Prior to their beginning training as Padawan learners, younglings would undergo a series of tests known as the Initiate Trials, as well as The Gathering, an important Jedi rite of passage that tested them on the frigid planet of Ilum.
Now let us look at example of a few canon people we see that are Force-sensitive but not recruited at a young age:
Anakin Skywalker - Found on Tatooine at about age 8-10. Qui-Gon Jinn wanted to train him as a Jedi due to his midi-chlorian count. However, Grand Master Yoda famously thought that Anakin "Was too old to begin his training". Anakin was unaware of his powers with the Force until being told about them by Qui-Gon.
Luke Skywalker - Lived on Tatooine with Aunt and Uncle until his late teenage years. He was just a farmer until being thrust into the rebellion by a series of events. Luke is unaware of any of his Force related powers until he is by Obi-Wan and begins a basic training during their travel to Alderaan.
Rey - An orphan on the planet Jakku with a similar story to Luke's. She in her late teens and unaware of any Force related abilities she might posses until they are exposed to her by another Force user.
No there is no age limit there. In the episode children of the force of clone wars: http://starwars.wikia.com/wiki/Children_of_the_Force there were babies of what semeed like only a few months age that were clearly force sensitive and exhibited telekinetic and other abilities already.
Despite their age they were already KNOWN to the jedi who had left them with their families for the time being until they were old enough to be taken in and trained to become new jedi.