Just to get the official rules in here:
The details of adding or removing these abilities during combat are sometimes misunderstood.
702.4. Double Strike
702.4a Double strike is a static ability that modifies the rules for the combat damage step. (See rule 510, “Combat Damage Step.”)
702.4b If at least one attacking or blocking creature has first strike (see rule 702.7) or double strike as the combat damage step begins,
the only creatures that assign combat damage in that step are those
with first strike or double strike. After that step, instead of
proceeding to the end of combat step, the phase gets a second combat
damage step. The only creatures that assign combat damage in that step
are the remaining attackers and blockers that had neither first strike
nor double strike as the first combat damage step began, as well as
the remaining attackers and blockers that currently have double
strike. After that step, the phase proceeds to the end of combat step.
702.4c Removing double strike from a creature during the first combat damage step will stop it from assigning combat damage in the second
combat damage step.
702.4d Giving double strike to a creature with first strike after it has already dealt combat damage in the first combat damage step will
allow the creature to assign combat damage in the second combat damage
step.
702.4e Multiple instances of double strike on the same creature are redundant.
Related:
702.7. First Strike
702.7a First strike is a static ability that modifies the rules for the combat damage step. (See rule 510, “Combat Damage Step.”)
702.7b If at least one attacking or blocking creature has first strike or double strike (see rule 702.4) as the combat damage step begins,
the only creatures that assign combat damage in that step are those
with first strike or double strike. After that step, instead of
proceeding to the end of combat step, the phase gets a second combat
damage step. The only creatures that assign combat damage in that step
are the remaining attackers and blockers that had neither first strike
nor double strike as the first combat damage step began, as well as
the remaining attackers and blockers that currently have double
strike. After that step, the phase proceeds to the end of combat step.
702.7c Giving first strike to a creature without it after combat damage has already been dealt in the first combat damage step won’t
prevent that creature from assigning combat damage in the second
combat damage step. Removing first strike from a creature after it has
already dealt combat damage in the first combat damage step won’t
allow it to also assign combat damage in the second combat damage step
(unless the creature has double strike).
702.7d Multiple instances of first strike on the same creature are redundant