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I know the common strong and weak acids but if a mineral acid like $\ce{H3BO3}$ springs up, I can't actually tell whether it is strong or weak.
Also can I justify in chemical terms why $\ce{H2SO4}$ is a strong acid but $\ce{H2SO3}$ a weak one?
Why is $\ce{HNO3}$ a strong acid whereas $\ce{HNO2}$ a weak one ?
http://catalog.flatworldknowledge.com/bookhub/2273?e=ball-ch12_s04
The above link shares a list of strong and weak acids but fails to state why one acid is weak while the other strong.
Is there a formula or way to determine whether an acid is a strong one or a weak one without conducting experiments in real?
Yes, there are ways to determine whether an acid is strong or weak without going into the lab yourself. That being said, someone must still go into the lab.
The most straightforward way would be calculating ΔG for an acid-base reaction. Remember, a negative delta G means that a reaction is spontaneous. Since strong acids have a Ka > 1, their ΔG values should be negative.
A "strong" acid simply means that when added to water, the acid will dissociate completely, therefore a strong acid is strong because it dissociates completely in aqueous solution while a weak acid is weak because it does not dissociate completely in aqueous solution. There are only 7 common inorganic strong acids so I would simply memorize them.
As far as why they are strong acids i.e. why they dissociate completely while other acids do not is a little more complicated and to be honest I am probably not the person to ask. I haven't looked into inorganic acids too extensively, though organic acids you must look at the stability of the conjugate base. A more stable conjugate base means the acid will more easily donate a proton. I do know for a fact that the conjugate base of sulfuric acid exhibits resonance which is a major stabilizing factor. However, based on the structure of sulfurous acid I would expect it to also exhibit resonance, though I could be wrong.
The acid dissociation constant Ka is determined experimentally. Strong acids have a Ka that is >> 1 while weak acids have a Ka that is << 1.
Hopefully I answered your question, but if not I'm sure someone with a little more inorganic chemistry experience will fill in the holes.
Short answer: In general, no; but oxyacids and polyprotic acids have an extensive literature. (See Wikipedia: Acid dissociation constant). Your second question, which is a "why" question is best answered on a case-by-case basis. Your question implies to me that you are not familiar with the quantum mechanical aspects of the dissociation of a protic acid in aqueous solution. I'm pretty sure they would almost immediately require much more advanced knowledge of quantum mechanics (specifically, DFT (and if you don't know what DFT is, then that proves my point)) than you have. In general, modern Physics and Chemistry (the hard physical sciences) aren't so good at answering "why?" questions. I encourage you to read (at least twice) the Wikipedia article Acid dissociation constant. It mentions a variety of contributing factors which influence dissociation. You have not only electrostatic effects (which in all but the most simple cases are quantum mechanical in origin (meaning require a QM explanation for all but the least sophisticated student - the least sophisticated students will accept half-baked explanations (like Pauling's 2 rules)) but steric effects, resonance effects, etc. QM is the basis for chemistry. Add statistical mechanics to that, and a "sufficiently powerful" computer, and you could recreate modern chemistry. By which I mean that fundamental "why?" questions can "really" only be addressed by QM. I'm thinking that that is not what you're looking for. (but, there it is.)
