Since my dad wanted to put a dimmer on his chandelier in the foyer, I installed a Lutron LED, CFL, Halogen compatible 3-way dimmer, since they're multiple switches (one 4-way switch, and two 3-way switches) hooked up to the chandelier, which had sixteen LED filament bulbs in each socket, and each of them were four watts each making 64w. So when I replaced one of the 3-way switches, with a 3-way dimmer switch, with the power shut off of course, after I installed a 3-way dimmer switch, turned the breaker back on where I was working, and tried it and all of a sudden, the bulbs in the fixture started to buzz like a fluorescent ballast at around 50% brightness, when the dimmer is all the way down. But very slight buzz in the dimmer. Is this normal? Because I did wire it correctly for 3-way operation.

The choppy waveform the dimmer outputs can cause electronic components to vibrate and hum.  Even incandescent lamps may put out a ringing sound when dimmed depending on the filament.  The buzz from LEDs and CFLs is sometimes louder and harsher.

Dimming LEDs lamps can be a hassle.  Sometimes they'll buzz, sometimes they'll flicker, and sometimes they won't dim low enough.  Performance can vary depending on the specific lamp/dimmer combination.  The number of lamps, and whether or not they're all the same type can have an effect as well. 

Overall the Philips WarmGlow are some of the best I've come across in all terms of no noise, no flicker, and good range, though even a couple of those haven't worked perfectly on all of my dimmers.

The dimmer generates steep voltage edges during normal operation.
The LED ballast needs RF filter, what is composed of capacitors, inductors.
The capacitors tend to be the ceramic ones for space reason. But the available ceramics tend to have strong piezoelectric effect and usually are directly soldered on the board as SMD's. The sharp edges cause them to vibrate, the rigit attachement to the board allows the board to act as a sound board and so transmit the vibration to the air around.
Plus the voltage edges form current spikes inthe inductors, which then vibrate their cores via the magnetostriction effect. Another source of noise.

Plus the low LED input power tends to make the dimmer operation erratic, when it takes few "tries" to turn ON the triac in each halfwave, generating way more than just a single voltage edge. These burst then make way loder noise than just the edges alone, so all the stuff becomes even louder. The last could be verified/eliminated by connecting a true resistor (an incandescent) parallel to the LED lamps (10W for 120V, 20W for 230V for LEDs up to 20W, then about 10W of incandescent for each 20W extra LED power).
If the noise disappear, you have the last problem. But be aware, such erratic behavior is usually accompanied by component overstress within the dimmer (the pot may burn out at high settings, the triac often operates above di/dt rating and dies soon as well), so it is not any bad idea to verify that.
The presence of these problems very strongly depends on the exact dimmer design and used internal components. There are new dimmers rated just for the LEDs (use a MOSFETs instead of a triac and cuts the sine on the trailing edge, so no high current spikes at all), but these usually are not usable for anything else...

I installed a Lutron LED, CFL, Halogen compatible 3-way dimmer …
But very slight buzz in the dimmer. Is this normal?

Lutron and others market a significant chunk of their dimmer lines as "CL".  A "CL" dimmer is an "incandescent dimmer" with tweaks or features tacked-on.  An "incandescent dimmer" is a forward phase chopper, a leading-edge phase-cutter.  So too a "CL dimmer" is a leading-edge phase-cutting dimmer.

("CL" dimmers have two added features which "just incandescent" dumb dimmers lack: range trimming and "kick start".  These features are good to have, yes even with plain hot wire incandescent lights.  A CL dimmer can defeat "pop on".  Marketroids will say pretentious hooey: "L in CL stands for LED; CL is ideal for LED".)

I discourage the purchase of "CL" dimmers for "CL" application.  My stance is a matter of principal: a job worth doing is worth doing right.  For electronic integrated "CL" dimmable lighting, an "ELV" dimmer is appropriate.

Electrical current profile / waveform / shape / phase-angle, of "CL" (CFL and LED) integrated electronic lamps, usually tends toward that of a capacitive (rather than inductive) load.  A leading-edge/forward-phase dimmer is inappropriate for a capacitive load.  The type of AC phase control which is better suited for capacitive loads, is "trailing-edge" "reverse-phase" chopping/cutting.  Marketting and copywriting departments label it "ELV dimmer".

These "ELV" dimmers are suitable for incandescent (resistive) loads, are NOT suitable for inductive loads ("magnetic" ballasts and transformers).

I am not an engineer, I am using words which I do not fully understand, and uttering statements which likely are improper.  I want to spread helpful information, not misinformation.  I welcome corrections.

comic relief:

http://www.lutron.com/en-US/Education-Training/Documents/LCE/LightSources/LED/Dimming%20LEDs%20-%20LFI_2012_v1.01.pdf

Dimming LED sources: what's working and what still needs fixing by Michael Poplawski (Pacific Northwest National Lab) and Ethan Biery (Lutron Electronics)

Page 66 out of 172 "Yes, it's dimmable!"

• "Dimmable" means "You can do something to reduce the light output"
• "Dimmable with most dimmers" means "Some of the dimmers we have in the lab reduce the light output"
• "Dimmable with standard dimmers" means "Some of the dimmers we bought at the local big-box store reduce light output"
• "Dimmable from 100-0%" means "The light output drops out below 10%"
• "Flicker-free" means "My boss can't see any flicker"

I have never heard of about any continuous dimming requirement in any code. That sounds to me like scam (they want to sell dimming solutions).
The two fixture set should be good enough where really high light levels are sometimes needed (one set for low level basic illumination, the rest to supplement the high level as really a minimalistic setup; of course putting both channels into the same fixtures is way higher quality installation).

True dimming is always less efficient than separate high and low channels. The high power equipment has significant loss contributor related to the power rating and present at all times, so compromises efficiency at low power setting. Dedicated low power sources may be less efficient at full rated power, but still significantly more efficient than dimmed high power channel when only the base light level is used.

Except some theatre effects, I do not see any practical reason why stepped light control won't be enough. On a few places I have installed a 3-way dimmable fluorescent ballasts along this concept, however I do not remember the last time I've ever used the medium setting. Very likely it was only during setup testing after installation and since then only the minimum an full power.