The Wikipedia article on "Stellar density" says the stellar density near the Sun is only 0.14 stars per cubic parsec. It suggests that the density in the central core and in globular clusters is about 500 times as great.
According to the List of nearest stars and brown dwarfs in Wikipedia , there are 61 stars within 15 light years of the Earth. Dividing 61 by the volume of sphere of this radius, we obtain 4.3e-3 stars per cubic light-year, or 0.15 stars per cubic parsec. Thanks to user31264 for providing this information, which is consistent with the information from the previous link.
According to the Wikipedia article on "apparent magnitude", the total integrated magnitude of the night sky as seen from Earth is -6.5. Making that 500 times as bright produces a total magnitude of about -13.2 (5 magnitudes is a factor of 100 in brightness). The maximum brightness of the full Moon is -12.92.
So even with 500 times as many stars in the sky, the total brightness would be only slightly greater than that of a full moon.
(This assumes that the average brightness of the core stars is similar to the average brightness out here in the Galactic suburbs.)
Parts of the core might be even denser than that.
(I've updated this with new information and deleted and old link whose numbers appear to have been incorrect.)
As it takes the sun about 250 million years (250 My) to orbit the galaxy, the proper motion of stars relative to the Sun will be the dominant effect of changes in the sky. The visible effects of the rotation will be far slower.
All stars move in the sky, some faster some slower, and in more or less random directions, not just moving around the galaxy. For example, Vega moves about 1 degree every 11,000 years. Around 12,000 BCE it was the pole star, and will be so again around 14,000 CE. Between now and then, other stars like Gamma Cephei and Iota Cephei will temporarily take the role of Polaris.
By 250 million years most stars will be far from their current position in the sky, but because of uncertainties it's impossible to say just where they'll be. For example, if the estimate of 250 My is wrong by just 1% (or 2.5 My), that means about 100 periods of Vega. Hence by that time Vega could be anywhere at all even if it stays in our general neighbourhood - which is certainly not guaranteed.
Galaxies move as well, but because they are much further away, their apparent position changes much slower than that of stars. It will be mainly our rotation around the galaxy that moves them in the sky.
Using telescopes we have already seen differences in the positions of the closer stars. In 10,000 years many changes will be visible to the naked eye. By the year 250 My, the sky won't look even remotely like the present.
Best Answer
Short answer, no.
The Sun's orbit is non-Keplarian; there are many perturbations and a general unevenness in the motion of the Sun around the Galactic centre. This is a result of non-uniform mass distributions, the galaxy not being a point mass, and the impact the relative motions of neighbour stars has on measuring. Thus, giving a particular eccentricity for the Sun is almost meaningless. For instance, it fluctuates up and down roughly $2.7$ times per orbit and it passes through high density regions which cause major perturbations. This creates instability in any average eccentricity.
Long answer, it is not impossible.
In theory, we could measure it. However, we have two rest frames; local and standard rest. The local rest frame refers to how we can take the average motion of stars within (say) $100~pc$ and use this average to compute our approximate orbital properties. The standard rest frame refers to us using Oort constants/properties and similar things in order to determine our more specific motion around the galaxy based on accelerational perturbations, etc. Both frames have their own advantages and both give slightly different values for our currently computed orbital characteristics. The problem lies with determining the relative weights each might contribute to an eccentricity value.
While the motion of the Sun may be non-Keplarian, we do know that the circular velocity is around $230{km\over s}$ and the peculiar velocity is on order $15{km\over s}$. This leads many astronomers to say that while measuring the eccentricity would be very hard and calculating it would be near impossible, they can say that it is most likely on the order of a few percent. Definitely less than $10\%$, but a value in the range of $e=0.02-0.08$ would be the most likely.