Yes, they ARE connected.
With two notable exceptions, volcanoes are associated with (a) tectonic plates splitting apart (Iceland and east central Africa come to mind) or (b) tectonic plates that are coming together (the Pacific Ring of Fire comes to mind). In the former case, magma is simply rising into an opening gap between crustal plates that are being pulled apart - like the mid-Atlantic Ridge. In the latter case, an over-ridden oceanic plate, loaded with water and chemical sediments, heats up as it goes deeper into an increasingly-hotter-with-depth mantle. Something called partial melting takes place: the lighter materials like silica and water and CO2 segregate from the down-going slab and float up - Mount St Helens in the Pacific Cascades, Sheveluch in Russian Kamchatka, and Mount Fuji in Japan are examples of these.
The notable exceptions are the volcanoes of the Hawai'ian Islands in the middle of the Pacific oceanic plate, and Reunion Island in the Indian Ocean. The generally accepted understanding for their existence is that a "hot spot" in the Mantle feeds up through a moving crust (the Pacific plate) and creates a string of volcanoes. In the Hawai'ian chain, the oldest are in the northwest, and the youngest are in the southeast on the Big Island. There's even a new one, called Loihi, that is forming on the ocean floor even farther southeast of the Big Island.
When we talk about moving tectonic plates, it's hard to come up with a reference point that everything is moving with respect to... Certainly the North American continent is moving westward over the Pacific and subsidiary plates, but Kamchatka is moving southeast over the same plate(s). If in fact there IS a "hot spot" in the middle of the Pacific plate - perhaps that is the one non-moving reference point on this entire planet.
With the increased recent activity around the "ring of fire", New Zealand, Japan and Gulf of California, is there an increased risk for earthquake in other areas of the ring of fire?
Geologic events never happen according to a regular clock - sometimes things are quiet around the Ring of Fire, sometimes several events happen in relatively close succession. There is no recognized relation between the huge Tohoku earthquake in Japan and the much earlier Christchurch, New Zealand event - they are too far apart in both space and time. THAT said, there have been several cases observed where a large earthquake has "lit up" distant volcanic or earthquake-prone areas. The large Denali fault earthquake of November 2002 apparently triggered swarms of small earthquake in Yellowstone, for instance. Nothing big happened, but there were a cluster of small earthquakes that correlate closely with the p-wave of the Denali event passing through.
The likelihood of other earthquakes around the Ring of Fire correlates much more closely with the rate of subduction - how fast the continental plate is over-riding and "smothering" the oceanic plate. This rate is much higher off the coast of Kamchatka, in eastern Russia for instance (about 8 cm/year), than the collision rate of the Pacific Northwest (moving only about 2.5 cm/year). For this reason the volcanoes in Kamchatka are historically much more active than those in the Cascades. In the 10 years that I've been receiving daily volcanic notices about Kamchatka, I'm at a loss to think of a time when a volcano in Kamchatka was not erupting. Whereas in the last century, here in the Pacific Northwest, we've only had Mount Lassen erupt (1915-17), then Mount St Helens in (1980-86).
Any plate motion will translate into earthquakes - the plates are scraping past each other - and the subduction (over-riding plate) earthquakes can be real doozies.
Slower tectonics translates to a quieter life: fewer earthquakes, fewer volcanoes.