How much of the suns radiation is infrared from compression heat, not fusion? Preface: Im a college level dropout, please dont stomp on my ignorance too hard.
As mass falls into a gravity well, it compresses and heats up. Even if fusion didnt exist/occur, in a body as large as our sun, I imagine that the resulting heat and infrared radiation output might even be significant.
Can this radiation output be quantified and might it be a noticable fraction of all the radiation from the sun?
Was the infrared compression-sourced radiation significant at the time the sun was born?
 A: An interesting question.  It's complicated because its hard to say how much infra red radiation "causes" radiation from the sun.  Those terms aren't typically what we use.  Or, if we do, its to point out that almost 100% of the energy we get from the sun is from infrared radiation off the upper layers.  Only the tiniest sliver of the energy of fusion at the core reaches us.  It mostly heats the outer layers, and we see the outer layers.
However, you talk a gravity well.  This is a bit easier to work with.  Both the effect of interstellar dust falling into the sun and the effects of fusion generate power which fuels the heat and eventual radiation off the surface of the sun.  We can quantify this.
To simplify things, I'm going to make some unreasonable simplifications.  We'll see how much energy would be released as matter fell from "infinity" (really far away) all the way to the center of the sun.  Now we know it all didn't collapse this far (that gets into the topic of black holes), but it makes for a nice upper bound to work with.
One of the convenient rules is that the gravitational potential (in J/kg -- how many joules of energy is released by moving a kilogram all the way to the center of the sun) is always one half of the square of the escape velocity -- $Potential = \frac{V_e^2}{2}$.  This is nice because we can just look up the escape velocity of our solar system (which is basically the escape velocity from the sun, since it's the main source of mass).  Doing it with Universal Newtonian gravity $F=\frac{GmM}{r^2}$ could be more of a pest because my simplification ends up dabbling around a singularity.
Escape velocity from the sun is around 617km/s.  Thus, our gravitational potential is 190.3445 GJ/kg.  Multiply this by the mass of the sun, which is rougly $2\cdot10^30 \text{kg}$, and we get $3.7\cdot10^41{J}$ of energy.  That's how much energy is available due to the gravity well pulling in all of the matter in the sun.  Since I'm making so many assumptions, I'm just going to round it off to $10^41 \text J$.  Let's not pretend I have earned any significant figures here!
So how much energy is that?  Off to one of my favorite tables on Wikipedia: Orders of Magnitude (energy)  (I am such a nerd!)

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*$2.276\cdot10^{41} J$ - Gravitational binding energy of the Sun

Hey, look!  If I had known better, I could have just looked up the number you wanted, rather than going through the calculations.... anyhow...

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*$1.2\cdot 10^{34} J$ - Total energy output of the Sun each year

*$2.276\cdot10^{41} J$ - Gravitational binding energy of the Sun

*$1.2\cdot 10^{44} J$ - Approximate lifetime energy output of the Sun.

So the gravitational energy of all of the mass the sun has captured is about equivalent to what it emits in 10 million years.  It is also equivalent to about 0.2% of the total energy it will emit in its lifetime.
As for whether it counts as "significant," that's really up to how you define that.  But its worth noting that the sun formed over 50 million years, and gravitational potential generated about 10 million years worth of solar radiation energy.
I am sure you'd have to do a more exacting calculation, but I think those rough figures might help!
A: The Sun is not currently getting smaller, it is getting bigger. So on the face of it, none of its radiative losses arise from gravitational contraction and heating.
Actually, it is a bit more subtle than that, because whilst the envelope of the Sun is expanding, the core of the Sun is contracting as the hydrogen is gradually turned into heavier helium. However, the percentage of the Sun's luminosity arising from processes other than nuclear fusion is very small, because that is the working definition of what a main sequence star is. According to the models of Siess et al. (2001), about 99.82% of the solar luminosity is attributable to nuclear fusion.
As for what luminosity arose from gravitational contraction before nuclear fusion started, well the answer is almost all of it, because there is no other source of luminosity. The protosun was much more luminous than the Sun is now, but that luminosity decreased as the protosun contracted and actually reached a minimum that was about a factor of 2 smaller than today's solar luminosity. NB: I am ignoring deuterium fusion, which very briefly adds a spike of fusion energy to the mix when the protosun is less than a million years old.
A: When the sun was forming, the heat from compression would have been the primary source of radiation. It was also the heat that initiated the fusion reactions at the core.  Today, very little compression is going on because the radiation from the core supports the weight of the gasses at most levels.  What little heat from compression that occurs would result from the shrinking associated with fact that the sun is losing about five million tons of mass per second in the form of radiation.  Any heat that does result from compression would be thoroughly mixed with heat from the core as they both work their way slowly to the surface. The radiation that we observe  is from incandescent gasses at the “surface”.
