calm bodily fabrications
Calm the bodily fabrications is to calm the breath. This is not by controlling it but having the intention but not the action to calm the breath. In many cases you don't need even this. It will calm down on it's own.
More on this see: page 35, Knowing and Seeing 4th Ed by Pa-Auk Sayadaw and Lecture 3 in Anapanasati: Mindfulness with Breathing - Unveiling the Secrets of Life by Ven Buddhadasa Bhikkhu
Preceding to trying to calm your your fabrication you should feel the whole body and breath otherwise this step will be premature. This is you should be able the feel the whole breath as well as you body. When breathing there are sensations that arise tied to breathing elsewhere in your body like expansion and contraction. You should be able to feel all this as well as any feeling even in remote areas of your body like your earlobes, toes, etc.
Question 1: Do they mean calming the body's physical tension, i.e. muscle tension or the mental sensation of tension in the body?
Calm the bodily fabrications is not the same as claiming physical tension thought this also seeds to be done.
You have to let go of physical and mental tension. These tensions manifest as an unpleasant sensation. You should look closely at these sensations as mere sensations without any aversion towards it. For a beginners you should try to divide and dicest the pain to see its constituent parts. If you can see it's arising and passing nature then perhaps you can concentrate on this. Initially even when you look at it the pain linger for a while. Not to worry just let go of this bring it to your meditation object. At later stages once you have look at its arising and passing nature the pain will just disappear. Do not wish or crave for this in which case this might not happen.
In summary the way to release tension is to try to see the tension or pain clearly and closely and once you see it its arising and passing nature.
Maha Saccaka Sutta your body should not be stained. So if your are fatigues maybe try lying posture. Also if you muscles are tensed loosen them. Sometimes you facial muscles may become tense. Best way to loosen them is to smile. Also smiling reduces tension and brings about pleasant or neutral sensations.
This occurs with the decrease of conceptualization.
Conceptualisation happens in verbal fabrication. Verbal fabrications is not explicitly mentioned in the Anapana Sutta but should be calmed before
calming the mental fabrication. This can be achived by keeping your attention focused on a object you are bringing down creation of new verbal fabrications. In the Abhidhamma there is 5 fold Jhana classification. In the Sutta there a 4 fold classification. In the Suttas the 1st and 2nd Jhana in the Abhidhamma system is amalgamated into one. The 1st Jhana has initial application and the 2nd Jhana is sustained application in the Abhidhamma system and in the Sutta system both initial and sustained application is considered in the 1st Jhana. Initial and sustained application is what enables thinking and conceptualisation. Once these fall away (Abhidhamma 3rd Jhana and Sutta 2nd Jhana) there is no conceptualisation or the mind does not have the ability to think or conceptualise. The definition of verbal fabrication is found in Cūla Vedalla Sutta and Aṭṭhaka,nāgara Sutta mention how verbal fabrication is stilled after the 1st Jhana.
Question 2: What do we mean by calming thoughts? Calming all thoughts? Gross thoughts? Trying to calm thoughts even though they will not ultimately subside?
This is to reduce verbal fabrication as discussed above. The main thing here is trying to reduce them. You have to have sustained effort to calm the fabrication by 1st anchoring the mind on an object then then continuously reviewing if it is with the object and re directing it to the object when it wanders away and perhaps even when it still is focused on the object followed by trying to see the object more clearly and closely. What you have to do is try and not develop aversion if you cannot sustain your attention and also not crave or get too attached to the object of being able to keep the mind with the object.
Breath meditation does have calming mental fabrication. This not claiming thoughts calming the perceptions and feelings, i.e., you do not perceive pleasant as agreeable and unpleasant as agreeable. In the Jhanic route this is in the 4th Jhana.
Also keep an eye on potential answers to this question: How does one calm one's bodily fabrications?
Question 3: Is clarity a kind of brightness of the object, or a greater focus? Is clarity a more detailed object? Is it a part of the object or the mind? How can it be recognized?
Once you keep trying to stay with the object followed by looking closely and clearly in more detail as per its construction and nature, as time progresses you will start seeing the object more clearly and in detail.
If the object or meditation is mind made then the clarity would of this object. If it is related to arising and passing then the clarity would be of this nature.
Looking closely is synonymous with Jhana in some definitions. See: What exactly is Jhana?
Final question: I seem to have tried to figure out all these things and meditate in a proper way. Yet, if I sit and try to meditate every day, and attempt to focus on the breathing, and I meditating in a proper way? I don't quite get what distinguishes proper and improper meditation, and whether I should meditate or not when possible. How long should it take to see either results or a shift in my meditation.
The worry you are having here might become a hindrance as you are not sure of your pratice. In Breath meditation in most cases you should just try look at things as they are and not control it. A few places you have to be active is to relax body and mind and also calm the fabrication.
What is proper meditation is if the meditation helps reduce your mental defilements. If your pratice feeds into it you are doing it wrong. See Pahāna Sutta which is a short description of letting go of the roots and Avijja Pahana Sutta 2 which concentrated on ignorance.
Also measuring the time to see results or longing for results lead further away from eradicating the mental defilements as this is a form of craving. The results will manifest itself when you have properly let gone (including searching for a result and measuring progress) and not creating any more conditioning / fabrication and past stock of conditioning / fabrication have been depleted.
Also see: Ānâpāna,sati Sutta