Have no experience with them. Would a good pro-quality DI box reduce buzzing and improve the general tone of my Telecaster into my Babyface?

It looks like the Babyface has a HiZ input, but the website doesn't say how hi a Z :) You may need a Zbox or something similar.
 
http://www.motu.com/products/guitar/zbox/body.html
 
If you have single coil PUs and they are not noiseless. you'll need to be careful about EMI. I prefer to go thru a guitar preamp set transparently and into line level. 

A DI, as noted above, is mainly for impedance matching. If you have an interface with an instrument input where the impedance of that input is too low, then you will lose high end. If you have any FX pedal with buffered bypass (like Boss pedals and many others) you can see if you hear any improvement with the pedal in the middle.

I do get quite a lot of EMI in my telecaster pickups, especially noticeable when using hi-gain or even "crunchy" amp sim settings. I realize Telecasters aren't really renowned for hi-gain sounds but still, I'm pretty sure when I've played them in the past through real amps, the buzz wasn't nearly so bad as it is going into my computer. 

a real amp,
would improve the sound of your guitar into your babyface.
 
your mileage will not vary.

Oh yeah I'm not denying that, but when you live in a situation in which miking up real amps isn't really an option, getting the best out of your DI/amp sim sound is paramount. 
 

A DI is a device mainly used to convert a guitars hi impediance out to a lo Z to send a clean signal to a mixing board. There will also be a hi Z output that goes to your amp on stage. In a studio one might use one in conjuction with a miked up amp for the engineer to have the clean sound. 
A real good DI like one made by Radial or Jensen might make anl improvment to the signal because they use better transformers ( Jensen) , A cheapo DI will make no difference comparied to your interfaces Hi Z. Might make it worse. 
There are many guitar pre amps that work without a speaker. My Blackstar as an example. 
The best tone is going to come from a real guitar pre amp. There is a sort of reaction between the PU's and the pre amp. I find plugging in my G&L directly to my interface is lifeless. 
There's certain types of guitar sounds that you can get away with that but not the type of music I play..I need the amp or the pre amp. 
 

I have gone back and forth on a DI box for a long time, but still never gotten one. As long as a signal is strong enough, a lot can be done with a signal once it gets inside the box before it ever hits an amp sim (and is almost mandatory for amp sims in my experience). I use a compressor/EQ combo prior to amp sims, and if I used single coils I would consider a gate in that chain as well.
 
EMI in particular takes some research of your environment to combat. Plugging in and throwing SPAN on an echoed track will let you walk around with the guitar and use it like an antenna to check the environment. Some things can be isolated (transformers, motors, ground loops, etc.), and others cannot. Kill the ones that can, and for the ones that cannot, throw another instance of SPAN in the chain with a gate/compressor/EQ between them. Tweak those to see if you get a clean sound you want (will high frequency roll-off), and then try the amp sim. This may be enough, but processing effects without this "EMI seek-and-destroy" mission first can get aggravating. (A funny experience of mine is my #1 EMI issue is a Yamaha mixer 3' from my interface... I track (in seriousness) with that off and through headphones.)
 
In actuality, the gate, compressor, EQ can be done inside the amp sim, but is much easier to isolate/control if used as a "common chain" for your setup, which is the real point of it. It also saves on "nasty surprises" when swapping presets within an amp sim since you do not have to change those settings each time (just pull those from the preset you switch to is easier).

In general, my experience between an interface HIZ and a DI box has yielded no real difference.  A DI will give you the ability to lift the ground - which may or may not do anything to help your situation.
 
 

if you want decent enough tones to commit to a project that you really care about,
get a real amp.
 
figure out which models have emulated outs and silent recording modes.
 
at the very least, get a guitar preamp.