Which collision energy at LHC is better for hunting 125 GeV Higgs, 7 TeV, 8 TeV or 14 TeV? Increasing collision energy in hadron collders doesn't always improve the abiilty to hunt down the Higgs. I know that if the Higgs mass is just above LEP exclusion, then even 7 TeV is too high to be optimum. However, I don't know the situation for a ~125 GeV Higgs.
If the Higgs mass is really around 125 GeV as recent data suggest, does it make sense for the LHC to increase the energy next year to 8 TeV, and does it make sense for the LHC to shutdown at the end of next year to prepare for 14 TeV run? Of course, increasing the collision energy generally benefits BSM searches, but I'm asking the question assuming only the Higgs is taken into consideration.
 A: First of all, the Higgs will be 5-sigma discovered after the 2012 data, even at 7 TeV.
Second of all, after the 2013-14 break, the LHC will probably restart at 13 TeV, not 14 TeV, in 2015.
Third of all, see

How many $fb^{-1}$ for the most likely $5\sigma$ 115 Gev Higgs at the 7 Tev LHC?

to notice that it doesn't hurt if you raise the energy. The required number of collisions drops approximately inversely proportionally – to one-half – if the energy is raised. Compare the 7 TeV and 8 TeV lines in the graph above to see that this is the rough rule, regardless of the Higgs mass. A higher energy and the same integrated luminosity means a higher potential for discoveries and exclusions, a higher expected confidence level.
You probably build on the fact that the low-energy LEP and Tevatron colliders were better in detecting things near 100 GeV. But that's not really because of their "advantage" of a lower energy. There's no advantage in having a lower energy, except for lower costs. It's because LEP and Tevatron collided different particles.
LEP made clear collisions where electrons and positrons could merge to pure energy – a virtual neutral particle including the Higgs – and cleanly decay. Even quarks and antiquarks in the proton-antiproton collisions at the Tevatron could do it, although there were many other quarks and gluons from the protons and antiprotons around.
However, the LHC may only make virtual Higgses by annihilating quarks with antiquarks, but antiquarks are "minority partons" in both protons. So new particles, including the Higgs, often come from gluon fusion - gluons inside the protons annihilate – and these processes are messier, less likely, and have relatively larger background. Moreover, gluons with higher energies are more frequent inside the protons than gluons with lower energies (parton distribution functions for gluons increases with energy), so the gluons themselves may be too energetic.
But the high energy of the LHC protons themselves is never the real problem.
A: The dominant production mechanism for the Standard Model Higgs is through gluon fusion, i.e. you have two gluons scattering with each other and producing a Higgs particle. The important backgrounds (collisions which look similar to Higgs bosons but are not concentrated at a particular mass) however are typically produced by quark-anti-quark scattering. 
If we have two quarks, antiquarks or gluons in a proton of each beam which carry a fraction $x_1$ or $x_2$ of the proton's energy respectively, we can -- to first order approximation -- produce a particle of mass $m$ where $m$ is given by:
$$
  m = \sqrt{s} \cdot \sqrt{x_1 \cdot x_2}
$$   
where $\sqrt{s}$ is the collision energy (e.g. 7,8,13 TeV). For $m = 125 GeV$ and assuming that both quarks/antiquarks/gluons have the same $x_i$, we get $x_{1,2} = 0.0179$ or $0.0096$ for $\sqrt{s} = $7 or $13 \mathrm{TeV}$. 
i.e. to produce a particle of a given mass, the average fraction of the momentum carried by the interacting constituents of the proton decreases with increasing collision energy.
The probability to get a quark, antiquark or gluon at a certain momentum fraction $x$ strongly depends on $x$. You can see this in the following graph:

(generated here: http://hepdata.cedar.ac.uk/pdf/pdf3.html )
You can clearly see that when going to smaller values of $x$, the probability to find a gluon (green line) increases much more rapidly (it literally goes 'through the roof' on this plot) than the probability to find a quark (black line) or antiquark (red line).

In addition to that, even with point-like particles like electrons and positrons, the cross section (interaction probability) for the production of a particle of a given mass increases with increasing energy of the collision because there are 'more possible velocities' for the produced particle (the 'phase space' is larger).
The Superconducting Super Collider in Texas with a center of mass energy of 40 TeV would have therefore been even better to produce Higgs particles, assuming that the luminosity (beam currents and beam focusing at the interaction points) would be comparable or better than the LHC.
