"re-rolling" a re-roll would be expressly forbidden by the rules, as it is in Warhammer 40K. When in doubt I often refer back to the Warhammer 40,000 sixth edition rules as a point of reference. You only ever get ONE re-roll in 40K and the result of the re-roll stands.
Under Fate Points, directly under the header "Limits on Assistance" on page 204, the Deathwatch rulebook reads: "Re-roll a failed test once. The results of the re-roll are final." This means you get just ONE re-roll and can't use another Fate Point to re-roll the test if the second roll fails. You eat the result even if it is worse than the first roll.
As for how many fate points a character can spend in a turn, the Deathwatch core says on that same page, that spending a Fate Point is a Free Action. On page 236 of the Deathwatch core, it explicitly states "A free action takes only a moment and requires no real effort by the character. Free Actions may be performed in addition to any other Actions on a character's Turn, and there is no formal limit to the number of Free Actions one character can take. The GM should use common sense to set reasonable limits on what can be done in a few seconds." This when combined with the information about fate points tells you explicitly that there is no limit to the number of fate points a character can spend in one given turn or round of combat, but it does explicitly say no more than one re-roll per Test.