Will grammar errors become correct after enough people use them for enough time?

We are most likely "doomed" in that these changes are more or less inevitable. But we are not "doomed" in the sense that the language is actually breaking or somehow falling apart. Languages have been around for thousands of years, evolving and changing; no language has ever evolved itself into a corner or created a construction that makes the language non-functional.

Many of the "correct" English words and constructions that we use today got their start as stupid-sounding "mistakes". Our case and gender systems have almost vanished completely, for example.

In fact, if I am not mistaken, a couple hundred years ago, "the only people who passed the test were Anna and I" would have been the pedantically correct version, and "Anna and me" would have been the sloppy "wrong" version; it used to be that the copula would have nominative case on both sides (e.g. "it is I", not "it is me"). Maybe that puts things into perspective a bit.

If you find certain things silly, there is nothing wrong with feeling that way and avoiding them — and not all variations become mainstream. But, it is pointless to try to make any significant effort to stop these changes. Mainly because it is inevitable, but also because you'd only be protecting a momentary instance of a thing that is constantly in flux.

(Edit: I should also add that I don't even necessarily agree with you in the case of lay and lie. Most people don't even know the correct paradigm for conjugating these verbs. I interpret the changes in popular usage to mean that these verbs are undergoing regularization; the old paradigm seems to be inherently confusing and I welcome the change: the language is fixing itself. I thought this might also be an interesting point of view to consider.)

Addendum based on the question's first edit: I recommend reading up on Standard Arabic to consider attempts to stop language change in its tracks. Because Muslims believe that the Arabic used in the Quran is holy, they have attempted to maintain this version of Arabic. This Standard Arabic is the only "official" Arabic, it is the only one they learn in schools, and it is the only one they write in. But, in day-to-day use, you can't stop language change. Instead, every region has evolved a distinct dialect (and these "dialects" really stretch the definition of that word to its breaking point). In many cases, they are not mutually intelligible (for example, Moroccan Arabic and Baghdad Arabic), and nobody in the Middle East actually speaks Standard Arabic as a first language. Most people have an imperfect command of it anyway except for the most highly educated. Most of the time, if speakers of different dialects want to communicate, they speak in a simplified hybrid of their own dialects and Standard Arabic (leaving off things like case marking that exists in SA). The Standard form has become different enough from their spoken language that they can't actually manage to follow the rules automatically.

This also means that if I want to learn useful Arabic, I have to learn Standard Arabic to read and for some TV programs, and I have to learn another language in order to actually communicate with the people around me. There is not much practical benefit to this system (but they aren't doing it for practical reasons).

If for some reason we decide to freeze English as it is now and make a concerted effort to maintain this form as it is, we will inevitably end up with the same messy diglossia situation that they experience now in the Middle East.

Addendum after second edit: I can't keep up with this moving target :)

You asked,

If being pedantic is slavish adherence to outdated rules in the face of actual and foregone changed reality, then when do we conclude that a change is a foregone conclusion?

My answer: change has always been a foregone conclusion. Trying to stop English from changing is like trying to stop the tide.

A possibly-relevant quote:

The problem with defending the purity of the English language is that English is about as pure as a cribhouse whore. We don't just borrow words; on occasion, English has pursued other languages down alleyways to beat them unconscious and rifle their pockets for new vocabulary.
- James Nicoll

We are not doomed. We've never been in a better position.

Just think of the fact that we can now ask a question like yours and immediately have people reading, answering and discussing it.

Think of the Wikipedias, all the online dictionaries, the blogs about languages, writing, etc. This is all new. For people who want it, there has never been a better moment to be prepared to read, write and speak correctly.

Yes, we're doomed. In fact, I don't think the types of errors you are pointing out aren't even worth worrying about, given that even the BBC can't seem to correctly choose between "to" and "too". "There's" (instead of "there are") seems to be used and accepted by more or less everybody these days. And don't get me started on its/it's, there/they're, etc.

Yes, we're doomed. Luckily, in almost all cases, it doesn't really matter.