Molecular cause of cramps, spasms and strengthening in muscles? (incl. intro to muscle contraction)
I think a lot of your questions try to split the hair; is this happening at the chemical or the histological level and I do get what you're asking, but you should know that the distinction is often not worth making. Pretty much whenever a neuron's involved, the interesting biology is multi-scale.


  Is the membrane continuous along these tubules, or does the tubule just end somewhere inside the muscle fiber?


The membranes are continuous.


  When the muscle is twitching... is this neurological of nature, or is it related to a molecular cause in the muscle itself?


Most things you'd call a muscle twitch are at the whole-muscle-group scale, involving the coordinated contraction of many individual motor units, so it's basically neurological.


  When the muscle is cramping... I'm almost certain this arises in the muscle. What causes it? A malfunction with regard to the calcium ions?


A muscle cramp is a colloquialism for a couple of things that are quite different from each other. Overall, as with the previous question, if someone's experiencing a muscle cramp that means it's a fairly macroscopic phenomenon and it likely involves a whole group of muscle filaments, so it's neurological. Most spasms and cramps are neurologically mediated. 

The connections with electrolyte balances (cramps from low sodium, potassium, magnesium, or calcium) also hint at the neurological basis because neurons act on each other (and on muscles) by forming or dissipating ion gradients. You may know that low dietary calcium can lead to muscle cramps; if this was relevant to the calcium release within the myocyte (from the sarcoplasmic reticulum) then the calcium-starved muscles wouldn't be expected to chronically contract (which requires calcium) but to chronically relax. 

That being said, there's a lot of room for feedback mechanisms. So, let's say a person experiences a muscle tear; the tear is small enough that it doesn't compromise the function of the entire muscle group. In this case it's adaptive for the local damage to 'signal' to the rest of the muscle group to initiate spasm so as to stabilize the damaged structures as they're repaired. In this scenario the local damage would 'inform' a neurological (and/or endocrine) response that actually effects the spasm.


  Lastly, and on a slightly different subject, what are the microlesions in the muscles that occur during strength training, and what is the overcompensation that happens?


Last time I was updated on this (not my bag), there were some large questions remaining. The tricks used by growing muscle to establish large, regular arrays of contractile machinery, that is organized over many spatial orders of magnitude, are poorly understood. There are structures that monitor the overall organization and that detect any large deformations. Regarding overcompensation, let's play through a generic scenario. A muscle cell is loaded too much and becomes physically damaged. It has to stop taking orders (stop responding to contraction/relaxation signals) and to initiate repairs. The overcompensation results because muscle cell's not really capable of knowing how large it was before the damage, so the safe amount of repairs to do is extra. Probably there are epigenetic processes that let a muscle cell 'count' the number of times its been greatly damaged and to scale-up the response appropriately.

If you just consider two likely sources of damage -- mechanical strain and lactic acidosis -- you can see that there are widely different mechanisms that would be required to detect the damage and to initiate repairs.