There are some general types of dishware and some specific ones (with special names), but usually specific ones still fall into some general type, so I will mention some specific ones, but definitely not all of them. And there are lots of cases when you wouldn't be able to tell two types apart for sure. It's OK, since sometimes we have to deal with strange things that designers create, and sometimes there is something from another culture that breaks the system, and so on.
Here's list of some common names:
тарелка is a flat plate used for main-course, usually 20 to 30 cm in diameter. There is also суповая тарелка or глубокая тарелка (adjective can be omitted) which is not so flat, but it should have flat parts on the top. So if you're looking at its side, it looks like this: –,___,–
блюдце is a smaller plate used for deserts, sandwiches, etc. блюдце is also put under a teacup or a coffee cup, it would then have design similar or complementary to the cup.
розетка is even smaller than блюдце, but may be deeper (getting close to proportions of миска) and is used for jam or honey
блюдо is a big plate (40+ cm I think) used to serve dishes for several people (a
cake, a whole bird, a big fish, appetizers, etc.)
Any of those can be round, square, hexagonal, etc. Name would depend on size and purpose.
миска is deep and is mostly used for soup (should look like this: |__| or like this: \__/). If it's for soup, it's almost never чашка, with an exception of bouillon cup, but it should have ear-like handle (or two) and look like this c|__|, then it's чашка для бульона or бульонная чашка, but you can still call it миска.
– салатник or салатная миска is same as миска, but bigger, and can be still called миска.
Difference between миска and тарелка is that тарелка is wider and shallower, but to establish an exact boundary is sometimes difficult. I'd say that if it has a brim, then it's probably тарелка, and if its height is half or more of its width it's probably миска.
чашка is a cup for drinking (usually for tea or coffee) and should have at least one handle, it's narrower and higer than миска or чашка для бульона. Looks like this: с|_| or like this: c\_/
кружка is basically the same as чашка, but bigger. I think, originally it was for mostly used for beer, but nowadays there are smaller ones (300-400 ml, not 500) and lots of people use those for coffee, tea and any other soft drinks.
Again, there is some gray area between чашка and кружка. I'd say that if it's less than 300 ml, it's чашка and otherwise it's кружка
стакан is a handless vessel, usually made of glass (almost same as glass in English). This category contains old-fashioned, rocks, highball and pilsner glasses (maybe more, these are what I'm aware of). Note that it can have different height and height-to-width ratio, but стакан never has a handle or a leg.
бокал is a glass on a leg (e.g. wine glass, hurricane glass, martini glass, etc.)
Two more glasses are рюмка and стопка. In my understanding they are same as бокал and стакан (respectively), but smaller (for shots).
бутылка is a glass or plastic bottle that liquids are sold in, it usually has a narrow neck (you can't put your hand inside). If you can, than it's банка, and if it's more than 3 liters, it's usually канистра. Except for huge bottles of whiskey or other beverage if they are mimicking smaller ones. All three of these have secure lids.
графин is a fancy glass bottle, usually with a not secure lid, or without any lid at all. There are also things like декантер (to let wine breathe) or штоф (not sure what exactly is the difference).
кувшин is bigger than графин, has wider neck, and usually has a handle.
поднос is a flat thing that you put plates on (in fast food places, or to bring to other room).
– коробка is a box. It usually has a lid, but usually it's not secure
контейнер is a box for storaging food. It usually has a secure lid
Now to your photos:
(1) Please kindly have a look at this photo. How do you call the dishes shown in the photo - the ones for soup and rice, the one for fish, and the one for berries?
Ones for soup and rice are definitely миска, one for fish is тарелка and one for berries is either мисочка (little bowl) or розетка. I'd use the latter.
(2) Please now have a look at this photo. What is the Russian name for these rectangular boxes - поднос, коробка, or what? And please have a look at the round dish in the upper left corner of the photo. How is this dish called in Russian?
Rectangular boxes don't have an official name in Russian. I would probably call it поднос even though you usually put plates with food on поднос and not food itself. If these actually are bento boxes (that have lids and are designed to carry around), then you could say it's коробка or контейнер. Round dish at the corner also falls into gray area somewhere between миска and тарелка. I'd say it's closer to тарелка as it doesn't look very deep.
(3) Now this photo. How do you call this red dish, the one whereon you see sushi?
That is тарелка. You might call it блюдо if it's set of sushi for a company, but even in this case, тарелка is also OK, as it doesn't look that big.
(4) Now this photo. How do you call these wooden sticks whereon the chicken pieces are nailed?
These are деревянные шампуры, adjective can be omitted. For smallers sticks that hold olives or other snacks there is word шпажки (derived from шпага which is a rapier).
(5) Now this photo. How is the big glass called? Графин or бутылка? And how are the small glasses called?
Big glass is definitely графин. Small glasses are either бокал or рюмка depending on the size, which is not exactly clear. Probably бокал
Note: The definitions I used seems to differ from formal definitions (e.g. wikipedia says that блюдце is thing to place a cup on, everything else is тарелка of different kind), but this is how the words are actually used in spoken language (at least around me).
If you need more types of dishware or more formal definitions, you can start from exploring this category in Wikipedia.
