Do spies get damage resistance when cloaked?

Solution 1:

As of the Gun Mettle Update, yes.

According to the wiki page on the update, the Spy will now receive 20% damage resistance while invisible regardless of the watch used. He will also receive reduced time on debuffs such as Jarate or Mad Milk while invisible.

As for the Dead Ringer, two changes have been made to it in regards to damage:

Changed: Initial attack that triggers feign death has its damage reduced by 50%

Changed: Damage resistance on triggering feign death scales over time. 65% to 20% over 3 seconds

Another change added in the Tough Break Update means that the damage resistance for the Dead Ringer has been further increased.

Changed: Initial damage reduction when feigning death increased to 75% from 50%