If you leave only GPS checked, nope, your location will not be able to be retrieved either using WiFi information or mobile network tower info. It will always try to use GPS to get your location when an app polls for it.
Yup, if you switch Location Services off, as long as the app is properly coded and 'reads' the state of this switch, it won't be able to get a location. Some weather apps will default to the last known location or if you set a fixed city instead of Current Location. The location method you selected is only active if the Location Services master switch is active. If not, all location methods are off.
For your maps question, nope. If you turn Location Services on, you can still use maps, but unless you select either the GPS only or GPS + WiFi + Cell Data, you won't get a high-accuracy pinpoint and most apps will block navigation until you activate GPS (there should be a warning message saying so and taking you to your location settings if you try to do it). Here if you turn location services off, it won't give you your location in maps, but the degree of accuracy of said location and the services available from the maps app will depend on which method you selected. Also remember, that now, even if you have Location Services on all the time, if no app is polling for location (you turn off all Weather app pollings, you're not actively using maps or navigation, you've turned off location history for Google, etc.), even if you selected to use GPS + WiFi + Cell Data, none of these are actually used (and therefore GPS is actually off) until an app polls for your location.
For your final comment, I agree. I do prefer the standalone GPS button, too. But in Android's defense, their management has been greatly improved so now it shouldn't affect your battery life all that much. You can test drive it, for instance, if you use your phone for one day with GPS + WiFi + Cell Data and then one day with WiFi + Cell Data only... as long as your daily usage remains somewhat the same, you shouldn't see that much of a difference in battery drain from location services.