Zeus's answer is great and comprehensive. I am just going to add some additional thoughts for some of the OPs specific questions. Some below are opinionated based on PRACTICAL REALITY and experience.
Avoid ProPhoto in Practice
I normally recommend NOT using ProPhoto. It is an ESPECIALLY bad choice for someone just starting out. And it is a particularly bad choice if your image destination is D65 (i.e. web or video) and not print (D50).
Reasons To Use ProPhoto
- You are going to print. Especially if the print process to be used is a high quality wide-gamut process using more than CMYK (i.e. hexachrome etc). Using ProPhoto in THIS instance allows you to remain in RGB mode to do your image editing (must use 16 bit per channel or more).
Reasons to NOT use ProPhoto
- Higher Delta E errors.
- Wider gamut than you can display.
- Impossible to accurately proof before print.
- Imaginary color primaries means you could inadvertently clip relative to output and not know it.
- Will still require transforms to another space.
- Converting to another space adds substantial workload to the flow.
- Requires multiple transforms to go from its D50 wp to the D65 used in web and video. More transforms means less accuracy, more losses, more problems.
- Unpredictable results because of all the above.
- Requires working in a higher bit depth.
- More predictable results can be had for most print outputs by converting to the CMYK colorspace (i.e. SWOP) and soft-proofing and delivering with the print profile instead of using an arbitrarily large RGB profile and crossing your fingers it might work.
- Provides no benefit in most scenarios.
- In other words, it adds complexity and needless work without actually providing a real benefit in most practical applications.
It's Not The Size It's How You Use It
The main reason people say use ProPhoto is a set of misconceptions. Here are some:
ProPhoto is a bigger gamut so it has more colors.
No, incorrect. The "number" of colors is defined by the BIT DEPTH. Technically, an 8 bit per channel sRGB image has up to 16.8 million colors. ProPhoto in 8 bit ALSO has up to 16.8 million colors. But those colors are spread over a LARGER AREA, so there is more DISTANCE between them. This does not result in greater fidelity, but it does result in greater errors (visibly as "banding").
ProPhoto won't clip your image.
Yes and No, but Not Usually Relevant. Yes, it is easier to clip in a smaller gamut space like sRGB. The next larger common space is Adobe98, and if you are clipping in sRGB, then Adobe98 might be the better choice for that image. But if you are going to web, you still need to get into sRGB, so you are going to need to adjust the image to fit into the sRGB space at some point.
The best way to not clip is to use an unbounded linear floating point working space.
Your Image Won't Lose Anything in ProPhoto
No, incorrect. You always have the potential to "lose" image data and fidelity. How much you lose depends on things like bit depth and how transforms to other spaces will be performed. When you use ProPhoto, you will have to make transforms to an output space, introducing errors for instance.
As mentioned above, using a space where the primaries are farther apart means that the available bits have more distance between values so each color is less accurately represented.
Also, since ProPhoto is a D50 whitepoint, you have to use the ICC PCS to convert to sRGB and a bradford chromatic adaptation. Going through the PCS can twist colors, clip, and cause other unpredictable results.
The better way to "not lose anything" is use SmartImage when opening RAW.
ProPhoto is better for HDR imagery
Again not true, it's more about bit depth. If you have very HDR imagery, bit depth and data type are important concerns. And again, the bigger the gamut, the MORE BITS you need to describe color values therein.
Side bar on bit depth
Not all 16 bit data containers are equal. There is integer, and there is float, and there is Photoshop.
Photoshop first, as it's a special case. PS 16 bit is actually 15 bit. That's still 35 trillion color values for RGB, but only 32768 "evenly spaced" levels per channel instead of 65536. This is enough for ProPhoto when gamma encoded, but not really for HDR imagery.
Other editors use all 16 bits, but Photoshop throws one bit away if you open those images in PS. Regular integer 16 bit is 65536 "evenly spaced" values per color. When I say evenly spaced, I mean in terms of numbers not gamma encoding. Integers are whole numbers 1,2,3,4 and no "fractions", i.e. decimals like 1.253
Another kind of 16 bit is 16 bit float. This is a floating point format, and DOES allow decimal values like 1.24345 the important different is this: in integer, each number is a whole number, and there nothing possible in between those whole numbers. With a float format, there is a LOT in between those whole number values, so you can "vary the distance" between color values in a more practical way than with integer math.
Floating point math allows working in LINEAR mode as opposed to a gamma encoded mode. This is actually more useful to prevent clipping and working with HDR than the colorspace.
Note that Photoshop is not so great for linear mode, but AfterEffects IS great for linear mode.
Pros use ProPhoto, so it must be best
No, Incorrect. Pros rarely use ProPhoto except for specific circumstances that may warrant it. This is especially true with current ADoughBee products that allow "Smart Images" when opening RAW, which is more ideal way to "not lose anything" than using ProPhoto.
Setting Working Space
OP's question on working space: You can set the working space in PS under colors, and you could set it to ProPhoto so you don't get that warning, but hopefully all here have sold you on the idea of NOT using ProPhoto for your application.
sRGB is the Standard for the Web
sRGB is the standard colorspace for the worldwide web. Even if you don't tag your images, browsers will assume they are sRGB. CSS4 is going to introduce other colorspaces to the web (IMO that's a bad idea but whatever) but even then sRGB will be the DEFAULT.
The implication is that you need to end up as sRGB for EVERYTHING. And sRGB is FINE as a working space if it is not clipping your images.
I work in Hollywood as an editor, VFX Supervisor, and Title Designer. We do most of our work in a LINEAR workspace, not a gamma encoded one. Photoshop does most things (8 and 16 bit) in a gamma encoded space, whcih is also how images are saved in jpeg, PNG, and TIFF etc.
Linear means NO gamma curve (curve power of 1.0). Linear requires floating point math (we use 32 bit float). The advantage of linear is that the math works the way light works in the real world. SO the math is simpler and also "more real".
Linear is virtually "unbounded" in that you can have colors much brighter than what would be maximum white in an integer image, and you can even have negative colors as are in some colorspaces (imaginary colors). this means that you essentially CAN'T clip.
Working in a linear floating point space is actually the IDEAL solution. Unfortunately Photoshop is kinda bad at it. Try Gimp if its interface doesnt drive you mad.
Adobe AfterEffects (AE) is surprisingly better than PS for linear (being driven by the film industry) but sadly Photoshop is stuck in its legacy ways. While AE is made for film/video, when I want to work on a still image in linear, I do it in AfterEffects.