A puppy born without a ridge is as much a Ridgeback as its siblings. It will be as healthy as any other puppy in its litter. Oftentimes, the ridgeless ridgeback (or slickback as many call them) has the best conformation, but because it doesn't contain the signature ridge it is disqualified from the conformation ring and from certain events. However, it may do obedience, rally, agility (not sure about agility, perhaps somebody else can confirm this) or whatever you wish. It should not however be bred and therefore, it will most probably come with a spay/neuter clause in the contract.
As for the genetics of the ridge, nobody really knows how the ridge is passed along. For instance, a breeder can go for years without having a ridgeless puppy and then suddenly unexpectedly one will appear.
It was thought for a while that ridgeless Ridgebacks did not get dermoid sinus, but that is actually a fallacy and examples have subsequently been cited where a DS has appeared on a dog without a ridge.
If you are interested in a good looking pet to have fun with the ridgeless will give you all the characteristics of the breed.
I am not sure what you mean by a partial ridge. If you mean a short ridge or a ridge with only one crown, then your puppy would not actually be disqualified since the only disqualification is the lack of a ridge. However, it would not make sense for you to try to show the puppy in conformation as it would probably be excused for "lack of type" but you could participate in all the other activities. It will most probably also come with a spay/neuter clause in the contract.
Puppies are born with/or without what they will have their entire life. In other words, if a puppy is born with offset crowns, the crowns will be offset for its entire life. Likewise, if it is born with a short ridge, I am afraid much as many of us would like it to happen, the ridge doesn't grow as the puppy grows.
Hope this helps.
Caroline