Do scientists get funding to replicate research? No, not usually.
Does replication still happen? Yes replication of a previous work is often step 0 of new research. It happens all the time.
Do replication results get published? No, not usually. Unless it's a clear rebuttal of a famous paper.
Does this matter? Debatable. Science proceeds slowly, and bad papers tend to be forgotten unless they're easily replicable. It just takes time.
Does this mean that we can't trust science? Absolutely not. You just can't trust individual papers, prima facie. This is something that scientists are supposed to learn as undergrads. Just because it's in a journal (even a "good" journal, like Science or Nature -- some would say especially if it's in one of those journals) doesn't mean it's right.
What does this tell us about armchair science? It's utterly useless. The reason that you have to be a professional scientist is not because professional scientists are smarter, but because they have a huge depth of knowledge about a particular field. Any working scientist can point out published, cited papers in their field that are total crap. But you can't tell that a paper might be crap unless you've spent years reading all of the literature in the field. Yet people on the internet persist in thinking that they can cherry-pick a single paper from arXiv or PubMed, and make sweeping conclusions about a field.
[1] indeed, the usual HN climate-change critics are in this thread, pointing at and yelling about how science is "broken".