Do I understand correctly that taxation-based government expenditures allow GDP to grow? Suppose (it's an example) GDP= $1 000 000, Marginal Propencity to Consume(MPC)=0.25, that borrowing money is an absolute taboo for our government and that said government wants to increase GDP.

What can the government do to increase GDP of its country? Well, it can increase government spendings. The expenditures multiplier is equal to 1.33, meaning that for each \$1 spent by the government the GDP increases by \$1.33. But with borrowing out of question our government will have to tax population first in order to raise the money. But additonal taxation will decrease our GDP. Will our expenditure multiplier be big enough to compensate for effects of our tax multiplier? Yes, because the absolute value of the tax multiplier will always be less than the absolute value of expenditure multiplier, no matter what MPC we will have (except if we will have MPC=1, but in this case we will get divizion by zero error), consequently meaning that EM-TM always >0. In our particular example we will lower GDP by 0.33 for each dollar that we taxed, but at the same time increase by 1.33 for each dollar that we spent, meaning that taxation and consequent spending of N dollars will give us +N dollars to GDP, 1.33x-0.33=x(1.33-0.33)=x*1=x

1.Is my understanding of given model correct?

2.If it's correct, then why don't government tax and spend tax money much much more actively than they are doing today? It looks like it would be a "cheat code" for unlimited exponential growth. Maybe there are other factors that prevent it, but that our simple model ignores?
In your comment you say you want the answer in context of the classic textbook model for goods market/output equilibrium.

Following the Blanchard macroeconomics textbook lets consider closed economy so the output will be given by:

$$Y=C+I+G$$

where $Y$ is output/income (they must be always equal), $C$ consumption, $I$ investment and $G$ gov. spending.

The consumption will be given by: 

$$C = c_0 + c_1(Y-T)$$

where $c_0$ is the part of consumption that does not depend on income and $c_1$ is marginal propensity to consume (MPC) and $T$ are taxes.

In this basic model investment is just fixed $I=\bar{I}$

Solution to the model (i.e. autonomous spending) based on the set up above is given by (the solution is derived by substituting $C$ in second equation for its place in first one and then solving for $Y$):

$$Y = \frac{1}{1-c_1}[c_0+ \bar{I} +G -c_1T ]$$

Now since you say that we take it for granted that government runs balanced budget $G=T$ because borrowing is "taboo" we can substitute the equality between taxes and gov. spending so we have:

$$Y = \frac{1}{1-c_1}[c_0+ \bar{I} +G -c_1G ] = \frac{1}{1-c_1}[c_0+ \bar{I} +(1-c_1)G ]$$

Now to clearly see the effect of G spending lets just set $c_0=\bar{I}=0$ (this is not necessary but it cleans the equation). So we have:

$$Y =  \frac{1}{1-c_1}(1-c_1)G = G$$

Hence there is no multiplier on Government spending if budget is balanced. So without debt government cannot increase output in any way (at least not in this simple model). 

The intuition is that of course if government just takes money from left pocket to the right there cannot be any increase in output. Since in this model all agents have the same MPC and hence all spending including the Gov. spending has the same multiplier. If you take 10 euros from consumers and just spend it as a government you are not creating any additional output. you are just reducing $C$ - reduction of which would also be affected by multiplier, by the same value you are increasing $G$ which is affected by the same multiplier. The effects must cancel each other out. Only if you would allow for heterogeneous agents where people have different MPC government could increase output in the short term by redistributing money from people with low MPC to people with high MPC, but the output cannot be raised "exponentially" and this would only be one time increase - i.e. increase in level or short run growth. I would recommend reading the full chapter 3 from Blanchard et all (2010). Macroeconomics an European perspective, that deals with this in greater detail.