Do English-language speakers care about grammar so much? For example when someone makes errors concerning mixed tenses does that bother readers so much?
Some readers won't be particularly bothered but others will be seriously put off. You may well slide a substantial number of mistakes past the average reader without them even noticing. But it's doubtful that you've gained anything in the process, and if for every ten readers who doesn't notice there's one who bad mouths your work to others because they did notice then you're at a net loss. That's not to say that you must adhere rigidly to grammar at all times - there can be valid narrative and stylistic reasons to bend or even break the "rules", but what it always comes back to is whether it's adding value to the story.
Deliberate deviations aside there's no reason not to strive to have the grammar as "correct" as possible, and if anything wanting to use those deliberate deviations makes it all the more important in the rest of the text. For example if you wanted to subtly hint at a character having English as a second language this would be one way to do it - but you lose that utility if the reader is unable to tell whether the "mistake" belongs to the character or the author.
Assuming you are talking about prose and not actually speaking, as long as the meaning of the sentence is absolutely clear, then grammar and syntax are less important.
But, grammar rules around tense and pronouns exists to make communication clear. So, specifically tensing, it is pretty important to get it right when writing, because there are patterns of language and sentence construction in English where the writer could be talking about future perceptions of things in the past. If the tensing was incorrect the message could likely be confusing.
Pronouns like he, she, they, them, etc., are also really important. It is not hard when writing in English about a group of more than two individuals and create pronoun confusion about whom a they, she, he in a sentence involving 3 or more people is referring to. The only what to avoid it by using proper grammar.
Given those statements, English is very flexible and doesn't some much have fixed rigid rules, but has patterns and guidelines of usage that can be ignored or twisted to produce terrific and meaningful sentences.
An example from George Orwell's Animal Farm: "ALL ANIMALS ARE EQUAL BUT SOME ANIMALS ARE MORE EQUAL THAN OTHERS"
The meaning of this sentence, in the context of the novella, is crystal clear, even though it has a big logical problem and a grammatical error, namely the modifier on equal. But Orwell the writer was speaking a higher truth, about how language is manipulated for political purposes to manipulate ideas.
But, if you are concerned about your command of English grammar and syntax, don't let it hold you back from writing. The rules of languages are small matters, and inconsequential, compared to the craft of writing and the art of story telling. As you learn the latter two, and the most important skills, the other will come of its own accord, and you can always hire an editor to clean it up when you are done.