What physics does occur at short distances in QED? Let us take the standard QED ($e^-, e^+, \gamma$) as a model of QFT and ask what is its "short-distance" physics? 
They say the UV infinities appear because we do not know the real physics of short distances and initially we introduce it wrong. OK, but after renormalizations, what physics does remain? Do we replace the unknown/wrong physics with certain/right one? Can anybody describe it without appealing to unphysical bare particles? Have we an idea about the real electron from QED? If so, why we cannot use it as the input to construct a reasonable theory from the very beginning?
P.S. Moderators, please do not close my questions before they are answered, let people answer.
 A: Lets start from experimental data. What is an electron, what do we know about an electron? It is too small to touch or see or smell. Everything we know about an electron comes from several levels of proxies. We end up measuring a track circle in a magnetic field and get e/m, consistently for different "electrons" and we do the millikan oil drop  and get e and then we can assign a mass to these manifestations consistently. 
That is all we have for the electron, it has a mass m_e measured and a charge e, measured.
Nature has been good to us and a working theory exists for QED. Mathematics is a tool, it can describe and predict measurements but it is not something that creates reality. Reality is what one measures . If the theory predicts, it does not matter if it goes into a yoga position to do so, as long as it can predict consistently. They want to call them bare and dressed mass? Fine. Who can measure anything more than that the measured mass is m_e and and the measured charge e? 
Better theories/computations  may come up, but to be better they should describe existing measurements and predict more and different ones, that QED cannot explain, for anybody to pay any attention. Or be as overwhelmingly economic and  elegant as the heliocentric is to the geocentric pov. QED works.
added: I want to give an example from real physics history that I heard from the horse's mouth back in the 1980s, of how succesful new methods of computation overwhelm tradition and sweep over reluctances once shown to successfully predict faster and accurately.
Back in the Manhattan day project, a physicist think tank had been set up with the best brains of the time to calculate crossections needed for making the bomb. Feynman was a junior member of the team. They gave the group a problem and a week later people reported the result of their independent calculations, parallel processing. Feynman said that one afternoon he was lying on his bed with his feet on the wall, when the Feynman diagram method came to him, whole ( he had eidetic memory so he probably saw it). He calculated the current problem and waited impatiently for the report of the others. When he gained confidence that his method was as good as the long drawn out s matrix calculations he started playing games with the team. He would get the result in an evening, tell them the next day what they would find, and it would take them the rest of the week to confirm. 
Of course Feynman diagrams were universally accepted after that.
I was reminded of this story when I listened to the talk of Nima Arkani-Hamed which he gave on the twistor revolution. He finds extremely cumbersome the Feynman diagrams method and is exploring a new one that gives the same results as the thousands of summed QCD feynman diagrams. I was amused, and am sure that Feynman would have been too, if he were still alive. 
If a new computational method is faster, sleeker and as predictive, it will be adopted as surely as God made little cabbages.
In my experimentalist's opinion of course.
A: If by "short distance physics" you mean "arbitrarily short distances" then the answer to your question is that your question is ill-defined. You cannot consistently describe arbitrarily small distances within QED.
Here is why.
QED (as opposed to QCD) has a Landau pole and thus is inconsistent if viewed as a "fundamental" theory. Like most other quantum field theories QED should be viewed as an effective field theory with its range of validity. You should not trust QED as an appropriate description of Nature at (or beyond) the Planck scale (or even at the GUT scale or at whichever new physics scale may lie between TeV and the GUT scale). 
A: Let us take standard fluid dynamics as a model of continuum media physics. It is obvious what is its "short-distance" physics isn't it? These are atoms and molecules. Let's apply your questions to this case -- suppose that we know nothing about atoms and molecules. in that case we can still use hydrodynamics, don't we?

Do we replace the unknown/wrong physics with certain/right one? 

What is "wrong" and "right" physics in a first place? If your hydrodynamics predicts physical phenomena in it's range of validity then I'm free to call it "right". Even if it says nothings about atoms. Anything else is a philosophy. 

Can anybody describe it without appealing to unphysical bare particles?

Can anybody describe hydrodynamics without appealing to "unphysical" continuum medium? Yes -- if one knows about atoms. If one doesn't know then one can try to speculate about underlying theory. But it doesn't render hydrodynamics useless and "wrong". 

Have we an idea about the real electron from QED?  Why we cannot use it as the input to construct a reasonable theory from the very beginning?

Have we an idea about "real" stream of water? It is made of atoms, right? So in this particular case there is no "real" stream of water at all.  That's why we cannot use it as the input to construct a reasonable theory from the very beginning.
A: Before attempting an Answer to this Question, I up-voted Anna V's and Daniel Grumiller's Answers. If we consider that "QED" includes all the calculational apparatus of an expansion as a series of divergent Feynman integrals, their regularization, resummation, and renormalization, then Daniel Grumiller's criticism of the question seems unavoidable.
If we consider "QED" to be defined more minimally, however, just by a deformed free field Hamiltonian or Lagrangian, without commitment to a particular elaboration of that definition, then it seems possible that there is a reasonable, analytic answer, but we have not so far constructed any such mathematical approach. In such a case, however, I suppose that any calculational treatment would nonetheless have to make contact with Anna V's request for engineering utility, which I further suppose requires a convergent series expansion of some sort on some finite domain from some different starting point. It seems unlikely, therefore, that we should start from the "bare particle" free field, since we already know that constructing a series centered on that starting point that is convergent in some sense requires extraordinary, time-consuming measures. To my knowledge, no-one has proved that every possible expansion of QED is everywhere divergent, but functional mathematics is not the relatively elementary playground of complex analytic functions, so such a proof may be possible.
With this prelude, I feel myself in a position to give partial answers to the sequence of subsidiary questions:

Do we replace the unknown/wrong physics with certain/right one? Maybe, maybe not. Can
  anybody describe it without appealing
  to unphysical bare particles? I think the bare particles thing is a problematic starting point, I think we have to start somewhere else. Have we
  an idea about the real electron from
  QED? If QED has an alternative series expansion that has some engineering utility, then the current expansion presumably has some relationship to the new expansion, in which case, yes. If so, why we cannot use it as
  the input to construct a reasonable
  theory from the very beginning? It's not necessarily the theory that's problematic, it may be the calculational approach. Just as one example, all quantized free fields already have something similar to the scalar field's property that a 2-point covariance function $C(x-y)$ diverges for $x\rightarrow y$, which makes free fields a questionable starting point.

Here, however, I acknowledge a fundamental difference from your approach as I understand it, Vladimir, that, to me, algebras of operator-valued distributions are a "better" starting point than particles, so that I have had to contort my answers to get near your starting point, a process which may have left the above near to nonsense, though I find the process instructive for my own rethinking of my ideas. For a long time I have thought that QED was itself a poor starting point for constructing mathematically consistent models, whether physical or not, but (for reasons unspoken here) I find myself more open to the idea as a result of attempting to answer this question.
A: The benefit of this genre of posts, is that they are challenging. There will be no progress if we speak only of what we were taught.
Wisdom is the result of testing what we know, and everyone has misconceptions, including myself ;) Honestly: 
on wikipedia: 
Renormalization: ...
 was first developed in quantum electrodynamics (QED) to make sense of infinite integrals in 
perturbation theory. ...
What infinities? I can not conceive of any infinite amount that can be attributed to any object in this universe.
And why not? Because it would take an infinite time or infinite energy or a null or infinite length.
For example what is the logic of integrating the EM field from r = 0 to infinity? Can be done mathematically, but physically is nonsensical because the short total lifetime of the particle (and universe).
The wavelength of a photon is limited by the size of the oscillator where it originated. Enormous (or miserably small) wavelengths imply correspondingly long (or short) time of interaction. 
Thus the application of cutoffs is natural and the right response.
As for perturbative methods, the calculation of a quantity by an infinite sum (not of energy, wavelength, etc) is one approach to another still unknown function. A situation that could raise such representation is the existence of feedback between two components. Any physical process responds within the short interaction time and do not involve computers nor have endless time to do the calculation. 

Can someone explain, please, because I am troubled with Vladimir comment: 'At least, Classical Electrodynamics predicts EMW when the electron is accelerated. So this phenomenon (radiation) occurs always. Now look at the Mott, Bhabha, Klein-Nishina formulas - they are the first Born approximations describing elastic processes. No radiation is obtained at'
