linuxsteamwine
All tutorials I see on running Steam games through WINE require Steam to be installed using WINE. I use the native version of Steam, as in the official Linux version.
Is there any way I can run Steam games through WINE if I didn't install Steam itself through PlayOnLinux or WINE?
My OS is Ubuntu 14.04 and my environment is GNOME 3.
There is no best way to do it, as such. Both Steam and Wine are ever-changing, and today's best method might break tomorrow*.
People have reported that Steam works with just standard Wine.
Install Wine 
from the Software Center (or click the shopping bag icon).
Right click the installer executable, go to Properties → Permissions and check Allow executing file as program, then right click it again and select Open with Wine Windows program loader.
If it turns out that doesn't work for you, you can try using PlayOnLinux.
Marco has detailed the instructions on how to get it working behind the link.
If it still doesn't work, you can give Crossover Games a go. Though it is number three on this list on purpose.
In general, it seems that Steam will work okay. You should try to run it using out-of-the-box Wine first. If you run into any trouble, feel free to ask a question under the wine tag at Ask Ubuntu.
Also see this site for Steam games that work with Wine (and, by extension, PlayOnLinux).
* Note: If you get it to work, updates to Wine will most probably never break it. But it may well be the Steam people are fighting off Wine compatibility. It may also be they are actively trying to make their program run well under Wine (games publishers have been known to do that).
The research I've done shows mixed findings. Some sources say Team Fortress 2 only has partial controller support for Xbox 360 and PS3 controllers on PC. The Team Fortress 2 wiki also only has setup guides for Xbox and PS3. The support pages for using a controller on Source do say that there is no Mac support for controllers in Valve games. According to another page, there are drivers you can install to use a controller on a Mac. Here are the support pages if you want to check them out.
Unfortunately, control support is limited to begin with and seeing as there little to no native support for Mac's and only minimal support on PC, I don't believe there is support for Linux. Sorry :(
Best Answer
Yes and no.
First of all: This is not directly related to any game, so you're better off at a Linux forum for help. I recommend you the most popular Linux subreddits on reddit.
1. No ...
... it's not possible in a very convenient way. And to say this at the beginning, you're really better off having a native Steam and a WINE Steam and switch between them when you need either of it.
2. Yes, you can!
Ok, there is a way to do it like you want it. There's a GitHub project called SteamBridge which lets you start WINE Steam games per native Steam. AFAIK you can also use Steam overlay from the native one.
The downside is that as it uses a custom Steam DLL, this clearly breaks the Steam TOS. So especially if you do competitive online-gaming do not use this as this may lead to a VAC ban.
Some users report that they use this a couple of time and never got problems with it, other state they got banned because of it.
You may try it, but I warned you.