When I download Wine to play Windows games, will I also be able to play those Windows games through Steam?
From my own experience I can confirm that Steam for Windows works great with recent versions of Wine.
By this you can run many naitve Windows games available through Steam using Wine or Play On Linux. Issues and games which won't run, or tweaks needed are best being queried from the Wine Application Database. It may also be worth to have a look at the Steam support forums for issues.
As a rule of thumb: if a game is listed as Platinum you will be able to play it without issues. A game rated as Gold will be playble just fine but may need some tweaks. All other ratings indiciate more or less severe issues.
Performance of games played through Wine varies considerably but many games will just perform fine with only a minor perfomance loss as compared to native Windows. This will also include 3D graphics which is supported through DriectX drivers in Wine.
Sadly however there is no guarantee that any single game you bought will actually run on your specific system. In case a Linux version is available through Steam you will probably better install this version.
It is also possible to connect your Steam account to both, native Steam for Ubuntu, and Steam for Windows run through Wine.
Steam for Windows under Wine works wonderfully. It lets me play Unturned and Terraria (the two main games I cannot play on linux) without issues (so far). Moreover, if the game you are looking at is a source engine game, such as Team Fortress 2 (which is, by the way, also available on Steam for Linux natively,) then you may run into some text rendering issues caused by a lack of native windows fonts. However, games that don't use said fonts (such as Times New Roman, Arial, Tahoma, Comic Sans MS, etc.) should work phenomenally.
However, as pointed out by a few people, if you are playing games which are on their own very resource taxing, it would be best to dual boot with windows (Yes, I spelled it like that because I don't believe windows deserves enough respect to capitalize it.)
There is some complexity but generally it is impossible to answer this question. Some games, yes, others, no. There are, however, lots of alternative ways to run Steam games on Linux.
A lot of games are now built for Linux, and you can view these in the Steam store by hovering over "Games" and selecting SteamOS + Linux - these games are all available as native Linux binaries. This probably won't be enough for most gamers, but it is at least something.
There is also a feature in Steam known as Home Streaming. This allows you to stream games from a Windows PC to your Linux PC without needing to have the game installed on the Linux PC. It's a bit flaky and doesn't always perform perfectly, but depending on the nature of the games you're playing it may well handle it just fine. You'll want to make sure both machines are on the same network and use the best connections available (that is to say if ethernet is available, use it).
From personal experience, I'm running Elite: Dangerous on an old laptop with an i5 and onboard 4000 series graphics, 4GB RAM and a fairly slow 5400RPM single platter hard drive, the game is of the same quality as it is on my xbox one.
You should really check the wine homepage. It will tell you how most games and other software work on various distributions using wine. Wine is able to make many pieces of Windows software work, sometimes with some workarounds. If you want an easier way to get Windows software working, check out Planonlinux. POL is a front for Wine, it does a lot of the configuration behind the scenes for you. If you want to pay for software you can also check out Crossover. It's made by the developers of Wine but it's more polished and has actual customer support.
Software that makes extensive use of accelerated 3D graphics, such as most mainstream Windows games, will not run well on Wine.
They may not run at all, or run with significantly degraded performance. Wine is good at running office or productivity applications but its limitations start becoming significant when pushing it with 3D gaming. You're unlikely to get a good experience with it.
On top of this, I have no idea how well Steam for Windows would run in Wine on Linux. You wouldn't be able to run Windows-only games with Steam for Linux, as has been pointed out.
If you are a heavy gamer and have a legal copy of Windows, dual-booting is the best solution for running heavy 3D games.
I agree with the first answer - Steam does work great with Linux, but some fonts and other graphical issues may crop up occasionally. By that I mean that the fonts on the "Updating" window appear more blocky and without anti-aliasing than the counterpart on Windows. Some games may not work correctly due to unimplemented functions in Wine - if you have a problem, first open the log, and find the DLL, and then go into winecfg and make it try to load the native libraries first. (There are online tutorials for doing this.)
Sometimes, other programs might require Microsoft .NET or other programs like Microsoft SilverLight or some other things which there are only Microsoft implementations of. In that case, there is little you can do if you only have Wine, but PlayOnLinux attempts to solve some of those problems by installing some of the programs you need such as Mono or Microsoft Fonts. It still doesn't work correctly sometimes.
There are some minor performance losses compared to Windows, and either way, I recommend that you install the proprietary nVidia or AMD/ATI graphics drivers. The FOSS versions don't have as much functionality compared to the closed-source version.
On another note, however, and yes, I know somebody else already posted this, there is a proprietary version of Wine called CrossOver, and it costs about $40USD for the cheapest option, and up to $60USD for the option with 12 months of online support and 1 month of phone support.
As my final point, I would like to make it clear that yes, I use Linux Mint 17.1 as my primary OS, but I still have Windows 7 installed so that I can use it if I have to, for running programs like the latest version of Office or things like Visual Studio. Windows is so cheap nowadays that you could probably pick up a version of Windows 8.1 for less than $100, so you can probably do that if absolutely necessary.