What are the odds of getting a gift from a Pokestop or gym?

From my observations of playing the game and spinning at least 2 Pokespots most days, as well as talking to my friends', the chance of receiving a gift appears to be about a 30% or 33% chance of receiving a gift when spinning a Pokespot or Gym. This chance appears to be identical for both Pokespots and Gyms.

Answering your other questions, also based on my observations:

• You can only receive at most 1 gift per spin
• You can obtain another gift from the same Pokespot when you spin it again at the normal rate
• Defending team for a gym makes no difference
• If a Pokespot gives or doesn't give a gift to you, that doesn't affect the chance of another person receiving it
• No bonuses for receiving multiple gifts in a row