Is it possible to lose synapses over time?
Regarding your question about losing synapses; yes, synapses are regularly lost in a process called Synaptic Pruning. From the Wikipedia article:


  A decrease in synapses is seen after adolescence reflecting synaptic pruning, and approximately 50% of neurons during development do not survive until adulthood. Pruning is influenced by environmental factors and is widely thought to represent learning.


Synaptic pruning is most significant during development due to "excessive" Neurogenesis that creates a large number of neurons, many of which will later be pruned. It's not excessive in a bad way, it's simply the case that more neurons are generated than necessary and thus pruning is needed to keep the connections efficient. There's no "measure twice, cut once", instead pruning is used to refine connections.

Synaptic Pruning really doesn't have anything to do with forgetting; Forgetting is natural and simply happens because connections (via Recall/Rehersal) are not made strongly enough to fully encode memories into Long Term Memory.

Note that Synaptic Pruning is not the cause for Alzheimer's or other age related degenerative disorders. From Wikipedia again:


  Synaptic pruning is classified separately from the regressive events seen during older ages. While developmental pruning is experience dependent, the deteriorating connections that are synonymous with old age are not. The stereotyped pruning can be related to the chiseling and molding of stone into a statue. Once the statue is complete, the weather will begin to erode the statue and this represents the experience independent deletion of connections.


Synaptic Pruning is a normal process of development and occurs in all individuals. Alzheimer's is a very visible disorder due to media focus, but not part of normal developmental/aging effects.

Instead I suggest you read more specifically about Alzheimer's disease, there's quite a bit of information on it.