Are axons in the brain weighted? Is it known whether the connection strength of synapses is important to the functioning of the brain or does just the binary existence of a synapse matter? Also, how widely do the strengths of synapses change, if any?
Is it known whether the connection strength of synapses is important
  to the functioning of the brain or does just the binary existence of a
  synapse matter?


I think it's safe to say that neuroscience, as a field, would stand behind this statement from neurobiologist David Sweatt, in his excellent 2003 (1st edition...2nd is probably even more current) book, Mechanisms of Memory, which he refers to as a "general hypothesis of memory storage":


  Memories are stored as alterations in the strength of of synaptic
  connections between neurons in the CNS.


So, yes, variations in synaptic strength is thought to be fundamental to the functioning of the human brain and, in fact, all nervous systems. And it need not be seen purely in terms of a learning &amp; memory paradigm. Any well-tuned network will need to have synaptic strengths set accordingly to produce a desired input-output function, so even if the strengths are hard-wired and unchangeable, they would still need to vary. It's important to note that synaptic strengths will vary over all types of synapses--classical excitatory and inhibitory synapses, and neuromodulatory synapses.

As of today, there appear to be 3,381 papers that are brought up when the search term "long term potentiation" (or "LTP", one of the most studied phenomena in synatpology) is put into PubMed as appearing in the title, and 935 with the words "long term depression"--and these are just two of the approaches to studying changes in synaptic strength. Habituation, sensitization, synaptic scaling, synaptic depression and facilitation (short time scale), and other mechanisms are also described. Looking at changes in synaptic strength is one of the biggest subcategories of cellular neuroscience research.


  Also, how widely do the strengths of synapses change, if any?


A classic LTP graph will typically display synaptic strength changes of about 200-250% of initial synaptic strength (or 1-1.5x more than you started with. If this were weightlifting and you started lifting 100 lbs, after you could lift 200 to 250 lbs). For an example of this displayed on a classic LTP style graph, see this graph.

And if you Google "LTP graph", you can quickly scan many such graphs. Most of these have similar amounts of potentiation (strengthening). But the range is likely more than just 2-2.5x strength, because these same synapses can be depressed from their baseline synaptic strength to below baseline, and in the intact animal's brain, the range of changes could (I'd guess) be larger.