Does Newtonian mechanics predict the bending of the course of light by objects with mass? $$F=G m_1 m_2/d^2$$
$$F=ma$$
$$a_g=Gm_{\rm other}/d^2$$
In Newtonian mechanics, the acceleration of object A toward object B is not dependent on the mass of object A but on the mass of object B and the distance between objects A and B.  Because the mass of object A does not affect its acceleration due to the gravitational influence of object B, does Newtonian mechanics predict that a massless particle (e.g. a photon) would be gravitationally affected by an object with mass?
Also, would it predict that an object with mass would not be gravitationally influence by an object without mass.  Would it also predict that two massless particles would have no gravitational influence on each other?
 A: The answer depends on whether light is a particle or a wave. If you imagine light is a particle of some mass travelling at speed c, then you get a Newtonian deflection of light, half of Einstein's value. This is discussed extensively in many places.
But if you think light is a wave, a wave doesn't fall, it only refracts. In order to get a wave of light to bend, you need the frequency of the light to change in different places. In Newtonian gravity, there is no change in the frequency of light waves due to their motion, because there is no coupling of gravity to electromagnetism, and there is no time-dilation which changes the frequency of waves.
So in the wave theory, it was expected that light would be unaffected by gravity. The two theories are reconciled in modern physics, since the time-dilation of gravity is the cause of both the deflection of light (by changing the frequency of light waves) and of the deflection of matter (by changing the frequency of matter waves).
A: First to the first part of the question:

does Newtonian mechanics predict that a massless particle (e.g. a
  photon) would be gravitationally affected by an object with mass?

With Newton's corspuscle-theory you get a bending of light, if we place a light bulb above a point mass the deflection of the rays would look like this:

Far away from the mass this is twice the amount of the bending we have around a Schwarzschild mass (in the strong field this simple factor of 2 is no longer valid and you have to use the full Schwarzschild- or Kerr-metric). For comparison, again the same scenario, but this time using relativity:

The mass is the same in both versions, with the difference that we have no point masses in relativity, so the diameter of the central mass is the event horizon instead of a point.
Because of the gravitational time dilation near the horizon, in the system of a stationary observer the rays will slow down and freeze when approaching the event horizon in the second image.
With Newton on the other hand, the light particles would be accelerated just like any other particle, and because Newton knows no speed limit, also exceed c after beeing sped up when travelling in direction of the mass (and again slowed down when receding from the mass - that's why they still turn around and escape if they don't directly hit the point mass, even when the escape velocity near the point mass is larger than c inside a radius of 2GM/c² in both theories).
The equations of motion and the code used in the animations and some other examples with rotating black holes can be found here and here.
The answear to the second part of the question

Also, would it predict that an object with mass would not be
  gravitationally influence by an object without mass. Would it also
  predict that two massless particles would have no gravitational
  influence on each other?

is no (it wouldn't predict that) in relativity (because energy and mass are equivalent, and also light rays have energy, see for example the Kugelblitz black hole made of light) and yes (they don't) for Newton's theory (then mass would be the only source of gravity).
A: You've got to be careful here to distinguish between Newtonian gravitational mass (gravitational "charge") and Newtonian inertial mass (measure of inertia).
Could there be, in Newtonian mechanics, a particle with non-zero inertial mass but no gravitational mass?  I think the answer is, in principle, yes.  It would simply be a particle that does not gravitate.
Could there be, in Newtonian mechanics, a particle with no inertial mass?  Such a particle would have zero momentum always (unless you allow for actually infinite speeds...).  Since, the particle's momentum is constant, by $\vec F = \frac{d\vec p}{dt}$, there is no, and can be no, force acting on the particle; the particle is a "ghost" cipher*.
So if, by photon, you mean a gravitationally massless particle then a Newtonian photon doesn't gravitate.
If, by photon, you mean an inertially massless particle, then a Newtonian photon is a cipher*.
*nonentity
A: Yes, Newtonian physics does predict light will be bent by massive objects. Newton mentions this possibility as Query 1 of his Optiks (1704):

Query 1. Do not Bodies act upon Light at a distance, and by their action bend its Rays, and is not this action (cæteris paribus) strongest at the least distance?

The bending predicted by Newtonian physics is half that predicted by GR (in the limit of a weak gravitational field), though I believe the quantitiative result was derived subsequent to Newton.
There's a nice, intuitive explanation of the reason for the difference between the Newtonian and Einsteinian results in Section 7.5.4 at https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-055j-the-art-of-approximation-in-science-and-engineering-spring-2008/readings/book.pdf  [Sanjoy Mahajan, course notes for The Art of Approximation in Science and Engineering, Spring 2008, MIT Open Courseware.]:

The most interesting constant is the 4 for general relativity, which is twice the Newtonian value because light moves at the speed of light. The extra bending is a consequence of Einstein’s theory of special relativity putting space and time on the same level. The theory of general relativity then formulates gravity in terms of the curvature of spacetime. Newton’s theory is the limit of general relativity that considers only time curvature; general relativity itself also calculates the space curvature. Since most objects move much slower than the speed of light, meaning that they travel much farther in time than in space, they feel mostly the time curvature. The Newtonian analysis is fine for those objects. Since light moves at the speed of light, it sees equal amounts of space and time curvature, so it bends twice as far as the Newtonian theory would predict.

Finally, it is worth noting that Eddington and Watson, when they were preparing for their famous experimental test of GR, were aware of this two-fold difference.  Hence their goal was not to determine if light was bent.  Rather, they assumed it would be bent, and were trying to determine if the degree of bending corresponded to Newtonian or Einsteinian physics.  That's part of what made the experiment so challenging — they weren't merely trying to show an effect, they were trying to quantitatively distinguish between two possible effects.
A: Yes, Newtonian physics does predict the bending of light by gravity. In fact, using Newtonian theory only, a geologist in 1783 noted that if the sun were 500 times larger in diameter but with the same density, light would not be able to escape it. That is, it would be a black hole.
A: What theory of light do you want to use? There doesn't seem to be any reasonable way to discuss Newtonian mechanics and photons; photons are innately quantum. We can't very reasonably discuss Newtonian gravity and classical electromagnetism, either, since Newton's gravitational law is not Lorentz-invariant, while classical electromagnetism manifestly is.
There was a time when people did calculate the effects of Newtonian gravity on light. Here is a paper from 1804 that does it. One approach is take an object of mass $m$ and initial velocity $\mathbf{v}$ moving past a star with some specified impact parameter. As $m \to 0$, the trajectory of the object converges, so we can take that as a solution for the path of light. (The trajectory it converges to is that of a "test particle" that has no gravitational influence of its own.) I don't know of any significant applications of such a theory.
Today we know that gravity is manifested as the curvature of spacetime, and so affects light, which travels along null geodesics in spacetime. The effects of the gravity of entire galaxies on light traveling through them can be very dramatic. This is what people study in the field of gravitational lensing.
A: Newton obviously knew that the mass of an object falling under the influence of Earth's gravity has no effect on its acceleration, i.e., all objects accelerate toward Earth at 32 ft/sec/sec regardless of their mass ("weight"). Therefore it follows that an object with no mass, such as a photon, would follow the same rule: it accelerates to earth at 32 ft/sec/sec. (The reason we don't notice this is that photons spend so little time between the object we see and our eyes due to their extreme speed.)
Newton obviously knew this, and logically concluded that photons from distant stars grazing the Sun's limb (edge) would "fall" just a bit towards the Sun as they passed by, resulting in a slightly curved trajectory.
A: It depends on your view of optics.
If one takes the (manifestly incorrect by the late 19th century) view that light is made out of tiny little massive particles with finite speed, or of the less-wrong theory that it is waves in some kind of massive medium, then yes Newtonian mechanics predicts gravitational lensing. 
Relativity, however, predicts that any causal influence will be lensed. This is why it is possible to talk about the paths of light rays without having to be terribly specific about how those rays actually work. This is completely alien to Galilean, including Newtonian, physics.
A: from Wikipedia:
"Henry Cavendish in 1784 (in an unpublished manuscript) and Johann Georg von Soldner in 1801 (published in 1804) had pointed out that Newtonian gravity predicts that starlight will bend around a massive object."
https://en.wikipedia.org/wiki/Tests_of_general_relativity#Deflection_of_light_by_the_Sun
See also:
"Newtonian gravitational deflection of light revisited"
https://arxiv.org/abs/physics/0508030
