Does the gravitational force of a massive, spherical body draw only towards its center of gravity, and to no other area?
Yes. This is exactly what Newton's shell theorem says, and also Gauss' law of gravitation.
This however only applies to bodies that truly are spherical. The Earth is not quite spherical. The most notable deviation is the Earth's equatorial bulge. (Even the tallest of mountains is significantly shorter than is the equatorial bulge.) This equatorial bulge makes for a measurable and useful discrepancy relative to what one expect from a spherical Earth.
One key benefit of this discrepancy is that it enables sun synchronous satellite to exist. The Earth's equatorial bulge results in a torque on satellite orbits. Place a satellite in low Earth orbit just right and the orbital plane will precess so as to keep the Sun angle constant over the course of a year. This is extremely useful for Earth-observing satellites (e.g., weather satellites and spy satellites).
To help illustrate: if you stand beside Mount Everest, there should be a pull towards the mountain even if its too slight to notice.
Not necessarily.
Newton considered (but rejected) using a mountain as a way to estimate the mass of the Earth. He thought the deflection toward the mountain would be too small to measure. This turned out to not be the case; instrumentation improved drastically after Newton's time. This drastic improvement can be attributed largely to Newton himself.
Newton's idea would have been correct if the Earth was a uniformly dense object. What Newton didn't know is what makes mountains mountains. Mountains, like icebergs, are big chunks of less dense material floating on a sea of more dense material.
One of the more famous results in early post-Newton physics was the Schiehallion experiment. This experiment used the Scottish mountain of Schiehallion to weigh the Earth. The results were amazingly accurate given the time.
After the fact, this was perhaps sheer dumb luck. Other experiments similar to the Schiehallion experiment didn't work. The Bouguer experiment with Mt. Chimborazo, conducted 30+ years prior to the Schiehallion experiment, didn't work. George Everest's work in the Himalaya, conducted 50+ years after the Schiehallion experiment, also didn't show the kind of deflection Maskelyne saw with Schiehallion.
Everest apparently was unaware of the Schiehallion experiment. Everest did see a deflection of the vertical, and he was confused by it. Others who were aware of the Schiehallion experiment checked Everest's results and were even more confused. The deflection observed by Everest was a factor of three smaller than expected. A number of other similar experiments didn't work.
There's a good reason for this. Continental crust is less dense than oceanic crust, which in turn is less dense than upper mantle rock. Mountains are in a sense chunks of light material floating on a sea of dense material. As is the case with icebergs, there's a whole lot more iceberg (or mountain) beneath sea level than there is above sea level. This tends to make the deflection of the vertical much less than that that result from a uniform density Earth. This is particularly so with large mountain chains such as the Andes, the Alps, and the Himalaya.
