Can an earthquake occurring between two plates, like the Pacific Plate and North American Plate cause any effect which would trigger an earthquake in a far away place like the boundary between the Eurasian and Indian Plates, as we know it is a mosaic of plates?
I guess that it is possible.
As mentioned by others, on top of static stress triggering, which is the effect of an earthquake in its vicinity (typically up to 2-3 rupture lengths away), another way by which an earthquake can trigger others is by dynamic stress triggering, which may have effects farther away but are hard to identify and measure.
So far, the only earthquake that seems to have demonstrated how dynamic triggering over long distances can take place is the 2012 Indian Ocean earthquakes. This earthquake had an unusual combination of characteristics: it was a very large strike-slip earthquake, it took place in oceanic crust (which allows surface waves to travel longer) and had a very large concentrated burst of Mw 8.4 - 8.5 with large slip over limited distance (I do not post sources since there are at least 3 apparently good studies with very different results, for example one study says that slip exceeded 80m locally (!)). These characteristics suggest that this earthquake was very efficient at radiating energy through the crust rather than towards the interior or through very fragmented crust.
So, what this earthquake showed, is that it is possible to speed up significant earthquakes (earthquakes that were almost ready to happen, but they were sped up up to 90 days). Still, it is only one case, the mechanism is not well understood, we are not sure why this earthquake seemed to have caused this phenomenon while other earthquakes with higher magnitudes did not show the same effects.
Concluding, while static stress triggering has only a fairly local effect, dynamic stress triggering may have a wider effect but we only have detected one such case, which could very well have been a coincidence or due to an unusual mechanism. So the answer to your question seems to be yes, but with doubts and in any case for very special earthquakes.
Without further qualification, sure they can. An enormous earthquake on one side of the world will shake the other side of the world, and if there's an incipient earthquake in a fault then that shaking could trigger it.
But, if you ask "is it common?", or "could an earthquake of X magnitude trigger another earthquake Y kilometers away?", then things get a lot more complex. My recollection is that I have heard of such triggering, but it was presented as a surprise, and certainly not a common occurrence.
Most times it does not happen, but there are many factors involved and in part it depends on your definition of triggering. As a general rule, earthquakes are considered to occur when the stress on one section of plate becomes too great and the plate slips, surges, or in some way moves to relieve the stress. The plates though are large, relatively solid objects, so slippage in one location to relieve stress is likely to increase it in other locations, some close, some far away. This increase in stress should tend to be spread out, and much less that what was built up and caused the quake as some of that energy is dissipate through the quake, but not all. If another section of that plate, or potentially even an unrelated plate was at or near its stress limit also, sure, that added energy could be the straw that breaks it loose.
The added energy from one quake is not really likely to be the cause per say of another large quake in a seemingly unrelated fault, but it could be the energy that causes it to pass its tipping point, and thus considered a trigger. The triggered quake could even be much large that the original quake. The second quake was going to occur anyway. The energy was already there and ready to be release, the jolt it got was simply the last push needed.
When two quakes occur, you will normally hear if you listen to reliable reports, statements that the two are not considered related. That is normally the case, but each time the seismologists look at the patterns and make that call based on their opinions. Sometimes they will decide they actually were related, but not often at least for large quakes.
I do not think that is possible. For one earthquake to trigger another earthquake at such large distances plates would need to be extremely rigid. Plates are far to ductile for stresses to be transported over such large distances.
In you specific example it seems even less likely. In your case stress would have to be transmitted over the entire pacific plate and then be transmitted over either the eurasian/pacific plate boundary or the indian/pacific plate boundary to be able to affect the indian/pacific plate boundary.