Do I make correct conclusions about international trade (and ways to increase GDP) from the expedenture approach formula for GDP? (Please correct me if my reasoning is wrong somewhere)

Net Exports is a part of GDP and consequently its increase or a decrease also causes GDP to increase or decrease, assuming everything else stays equal. Higher import means lower Net Exports, while higher export means higher Net Exports.

Let's take two countries, country A and country C. Let's also assume that all internatinal trade happens between these two countries. If country C will increase its exports to country A, then GDP of country C, assuming everything else holds equal, will increase due to increased Net Exports part of its GDP equation. While for country A its GDP will decrease due to its decreased Net Exports.

From this it seems logical to conclude two things:

1.International trade is a zero-sum game from point of view of increasing/decreasing GDP. 

2.If we want to increase our GDP, then among other things we must consider decreasing our import and increasing our export. For an example, we could outright ban some of foreign goods or raise tariffs, thus forcing our population to decrease consumption of foreign goods, this way increasing our GDP.

The only problem I see so far is that in the real world such policies can lead to trade wars, with other nations banning goods from our country and rasing tariffs on our goods, thus hurting our exports.
Your conclusion would be correct in pure endowment economy where there is no production $Y(L) =0$ (assuming only labor as factor of production) and everyone has the same endowments. In an endowment economy where there is no production and GDP is just dropped on both home and foreign country trade is indeed a zero sum game as $M$ subtracts from your endowment since $Y=C+I+G+X-M$ (Although its worth noting that in the long run trade must balance $X-M=0$ for reasons relating to exchange rate, savings, investments and movements of factors of production - but since it is not directly related to your question I wont go in depth into that) 

However, real life economies are not pure endowment economies. Production and output $Y$ depend on how efficiently we can produce output. In real life trade allows countries to produce more i.e. have higher $Y$ and thus GDP than without trade.

Consider simple Ricardian model where you have 2 countries utopia and neutopia. Both have 100 labor $L$ avaiable. Assume there are two products wine $w$ and cloth $c$. Utopia can produce 1 unit of $w$ for 2 units of labor and 1 unit of $c$ for 1 unit of labor and neutopia $1w$ for $1L$ and $1c$ for 2$L$. Now if they dont trade with each other and if they both split their labor supply equally between wine and cloth Utopia gets to consume $25w$ and $50c$ and neutopia $50w$ and $25c$. Now if instead of autarky they decide to trade and Utopia will specialize only in cloth it can produce $100c$ and Neutopia if it specializes in wine can produce $100w$. Now both countries can trade lets say 50 units of wine for 50 units of cloth and now both countries have $50w$ and $50c$. Both countries expand their production possibility frontiers by engaging in trade and trade is not zero sum as both countries are better off.

Hence, as a result of comparative advantage $Y$(trade) > $Y(autarky)$. So once you allow in the identity $Y=C+I+G+X-M$ the output $Y$ to be increasing function of country's  PPF its not correct to say that trade is zero sum game.

Of course, in a classroom setting its fine to apply national identity just to some pure endowment economy where your zero-sum conclusion about trade is correct. Or you could even allow for some production function but assume all countries have exactly same production function and exactly same resources endowments and there can be no economies of scale etc. where there is no reason for trade. But you are making a mistake by using a simple identity that ignores comparative advantage, factor endowments etc. and just making conclusions about international trade in real world.