Are Durins II through VII reincarnations of Durin I?
That's what the Dwarves believe:
There [Durin I] lived so long that he was known far and wide as Durin the Deathless. Yet in the end he died before the Elder Days had passed, and his tomb was in Khazad-dûm; but his line never failed, and five times an heir was born in his House so like to his Forefather that he received the name of Durin. He was indeed held by the Dwarves to be the Deathless that returned; for they have many strange tales and beliefs concerning themselves and their fate in the world.
Appendix A: "Annals of the Kings and Rulers" III "Durin's Folk"
A longer version of this text is included in History of Middle Earth XII (relevant bits emphasized):
For the Dwarves asserted that the spirits of the Seven Fathers of their races were from time to time reborn in their kindreds. This was notably the case in the race of the Longbeards whose ultimate forefather was called Durin, a name which was taken at intervals by one of his descendants, but by no others but those in a direct line of descent from Durin I. Durin I, eldest of the Fathers, 'awoke' far back in the First Age (it is supposed, soon after the awakening of Men), but in the Second Age several other Durins had appeared as Kings of the Longbeards (Anfangrim). In the Third Age Durin VI was slain by a Balrog in 1980. It was prophesied (by the Dwarves), when Dáin Ironfoot took the kingship in Third Age 2941 (after the Battle of Five Armies), that in his direct line there would one day appear a Durin VII - but he would be the last. Of these Durins the Dwarves reported that they retained memory of their former lives as Kings, as real, and yet naturally as incomplete, as if they had been consecutive years of life in one person.
History of Middle Earth XII: "The Peoples of Middle Earth" Chapter XIII: "Last Writings" Of Glorfindel, Círdan, and other matters
Whether or not this is actually reincarnation, as it was with Glorfindel, rather than a religious belief, is left ambiguous in the text.
In fairness, it's not impossible; Dwarves are a species very much unlike the Elves and Men, in that they were created by the Vala Aulë rather than by Ilùvatar, so the normal rules don't exactly apply to them1. However, I'm unaware of any textual evidence that explicitly confirms or denies if Durin is actually reincarnated.
How do the Dwarves know who to name "Durin"?
The quote above indicates that the Dwarf in question somehow resembles Durin, though what this means precisely is unclear and not discussed in the text. However, it's not necessarily a contradiction to say that they get named "Durin" later in life; Dwarf naming is a little special in that the name we associate with them is not their real name:
Gimli's own name, however, and the names of all his kin, are of Northern (Mannish) origin. Their own secret and 'inner' names, their true names, the Dwarves have never revealed to any one of alien race. Not even on their tombs do they inscribe them.
Appendix F I: "The Languages and Peoples of the Third Age"
"Durin" is most definitely an "outer" name; it's Old Norse, which is the "translation" Tolkien used for his Mannish names, taken from Völuspá. Presumably a Dwarf's "inner" name would be in Khuzdul, a language used very rarely in the text but which is wholly unlike the Mannish tongue.
When this "outer" name is chosen is unknown, but we can easily imagine that it's not until the Dwarf comes of age; before that time, there's no need to be interacting with non-Dwarves. That leaves plenty of time for Durin-like characteristics to be discovered.
Additionally, History of Middle Earth says that a Dwarf's "outer" name can change if the need arises:
The adopted names could be and sometimes were changed - usually in consequence of some event, such as the migration of either the Dwarves or their friends that separated them.
History of Middle Earth XII: "The Peoples of Middle Earth" Part 2: "Late Writings" Chapter X: "Of Dwarves and Men"
Presumably, Dwarf parents could give their little Dwarfling an 'outer' name at birth (or any time after it), and then change it to "Durin" whenever it became apparent that they were the reincarnation of Durin I.
1 It's the belief of the Dwarves that Aulë takes their souls to the Halls of Mandos:
For they say that Aulë the Maker, whom they call Mahal, cares for them, and gathers them to Mandos in halls set apart;
The Silmarillion Part III: Quenta Silmarillion Chapter 2: "Of Aulë and Yavanna"
This is the same place Elvish souls go, and they can be reincarnated after a while (with very few exceptions), so who's to say whether this is true or not?