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How could an earth sized and broadly / liberally earth-like planet have plate tectonics but still experience zero or weak / little earthquakes? How can I have the weakest earthquakes possible while maintaining an environment that is habitable for life?
Is it even possible?
Yes, under the condition that the crust of the planet is not yet solid. But that's hardly compatible with life as we know it.
Else it won't be possible to escape the following loop:
Lubrication of the rock layers may reduce the magnitudo of the earthquakes, but still the rocks would break.
Magnetic plates.
Earths crust is made mostly of siliceous minerals, aluminum and magnesium. But it is not hard to imagine plates that were more purely metallic and so were magnets in and of themselves. These could be big plates of iron, permanently magnetized. Or they could be conductive (iron, or aluminum or any conductive metal) and so be inductively magnetized by the larger electromagnetic effects in the molten core.
Two magnets aligned S to N will stick together and in essence become one large magnet. Plates like this will have stuck together long ago and for all purposes act as one plate.
However plates aligned N to N or S to S will repel. We can assume all remaining plates that are single plates are now so aligned relative to each other. The repulsion becomes stronger as the plates draw closer. This serves as an invisible bumper preventing any sudden energetic interaction. Imagine sliding one magnet towards another across a table - the other magnets are pushed aside or along in front. These magnetic crustal plates will not slide under or over each other but push other plates aside - or along in front. There will be no opportunity for a gradual build up of energy and a sudden release.
I could imagine all these magnetic plates moving along in a stately armada, maintaining position relative to one another, gradually circumnavigating the globe.
Notable: if a magnetic plate is somehow rotated such that it is now south to north with another plate, then you would get an earthquake and 1 plate where before there were two.
Presumably if it either had very soft tectonic plates or if they were well lubricated. Earthquakes occur if, during subduction (technical term for one plate being shoved under another), one plate gets caught, it can bend like a giant spring. Because of this, when it 'snaps' back, it releases an enormous amount of energy (i.e. earthquakes). So, if your tectonic plates are sufficiently soft and/or lubed (water can do wonders here - fraking is another way to do it) then it's conceivable that you could have plate tectonics without much in the way of earthquakes (not to say that they won't happen, but they'll be far less likely)
Consider that we only know of the existence of tectonic plates because there are earthquakes, volcanic activity, or other geothermal related activities. Presumably there may be many such tectonic plates beneath our feet which have long since been inactive and simply move with the other plate.
So in this regard, yes and no. Yes in that there may have once been tectonic plates, but no in the sense that cracks in the crust which do not slide past each other or push against one another may as well not even exist in any real capacity.
The earth too in theory will eventually arrive at the point where the crust is too thick for there to be any plate tectonic movement, but certainly not in the near distant future.
Your question may be divided into three parts.
The Plate tectonics but no earthquakes part has already some good answers, I will focus on the other two parts.
Experiencing zero or weak / little earthquakes
I experience weak earthquakes every day, when the next freight train passes within 50m of my flat. The several natural earthquakes a month in this region are to week to distinguish them from the footsteps of my neighbor. If there were no stories about it I would not know that natural earthquakes exist. This is not true for every place on this planet, but it should be for some of them.
Science can only provide probabilities for the next earthquake. Theoretically it is possible that whole generations over the whole planet do not experience any not-weak earthquake in their lifetime.
So in a given time in a given place there are no stronger than weak earthquakes even on earth, so it is possible.
Maintaining an environment habitable for life without earthquakes
As I understand the question, you try to maintain an ecosphere without regeneration of critical elements through subduction and volcanoes.
The answer is easy, it depends on your planets design. Elements like e.g. iron, hydrogen, oxygen and nitrogen have to be somehow introduced to the ecosphere and/or regenerated from deposits like sediments. Introduction might be accomplished through meteorite impacts or better some space dust (we don't want more earthquakes). If you need to regenerate more than through the normal biogeochemical cycles you might need some natural or artificial method to get sediments from the ocean floor and put them on higher ground like the bear-salmon-phosphor-pump, a combination of shallow water and strong seasonal winds (sadly no tides because of the earthquakes) or mangroves everywhere. I expect biology to adapt to smaller amounts of problematic elements, at least for some time, too, like the amazonas rainforest.
I could try to calculate how many years this ecosystem would be in a more or less stable flow equilibrium (meaning: individual inhabitants don't experience a change), but there are too many unknown variables here and even the research on this problem looks more like our nitrogen comes for several billion years out of plate tectonics and volcanoes so mars and venus are loosers than how long it would last without or how much it changed in the last million years. Just enter whatever time you need between some hundred and some billion years or put down some more of these meteorites before the time anyone can remember (because of the earthquakes of course).
I hope this helps.
