When do I need to use passive voice and where it's better to stick with active?
There seem to be two ways to say basically the same thing. See the following examples:
1) The column added to the report;
2) The column was added to the report;

The first issue is that whenever I use the second option Grammarly tells me about "Passive Voice misuse".
The second is that - how do I distinguish whether the first option actually means "The column [itself] added [something] to the report" or "The column [was] added to the report [by someone]".
a) Is the second problem solved solely by relying on the context?
b) How does Google Translate may know how to translate properly (meaning it probably can't use the context as people do)?

Question

When do I need to use passive voice and where it's better to stick with active?
There seem to be two ways to say basically the same thing. See the following examples:
1) The column added to the report;
2) The column was added to the report;

The first issue is that whenever I use the second option Grammarly tells me about "Passive Voice misuse". 
The second is that - how do I distinguish whether the first option actually means "The column [itself] added [something] to the report" or "The column [was] added to the report [by someone]". 
a) Is the second problem solved solely by relying on the context? 
b) How does Google Translate may know how to translate properly (meaning it probably can't use the context as people do)?

Answer

Vitaly, Grammarly has its uses, but there are many issues with it, so don't rely on it too much.  It does not even pick up incorrect spelling sometimes!! Rather ask us on this forum!!! :)

Firstly, if you use the passive voice, sentences can easily become clunky and clumsy and then sentence construction can go completely awry!!  .  Secondly, it sounds more natural to use the active voice.  However, sometimes you may wish to emphasize a word by using the passive voice, and that is perfectly acceptable.  Just don't use it much, that's all.

Answer

About your examples: we would need to see the whole sentence; there is not enough information in these fragments to tell if they are correct or incorrect.

About using the Passive/Active Voice: Use the Passive Voice if there is a good reason not to include the agent of the verb. Some common uses of the passive voice include 1) science 2) official (e.g. government/police) statements.

For example:
1) "The biological samples were stored at 0ºC"
2) "The budget was announced this morning"

In my examples 1) and 2) we often use the Passive Voice in science or in official language, because the process is important, but WHO did it is not important.

Some stylists recommend that you do NOT use the Passive Voice too much in ordinary writing because a) it is not clear who performed the action, so may seem like b) you are not giving the reader enough information, and c) it sounds too much like scientific/official language.