Can you still distinguish yourself at a second-tier grad school?

First, I think it is important to recognize that the definitions of "1st tier", "2nd tier", etc. are not so universally defined. World or national rankings of universities are substantially based on the opinions of department heads. The differences between the #1 school and the #20 school may not be that large.

Professors also move from "2nd tier" to "1st tier" schools all the time based on demonstrated research output. Therefore, it is possible to improve your value to hiring committees with more productivity.

Now to analyze your 3 options (which I think are fairly comprehensive): I think all 3 have some impact.

1. Departments often hire faculty with new expertise, meaning that no one at the department is an expert in that subfield. If they are not expert enough to evaluate individual papers directly, they may use as a proxy the quality of the journals which papers are published in, or the ranking of a school at which the student graduated from. However, I think this has the least impact because the quality of an individuals' research publications can usually be assessed separately from the ranking of their alma mater.
2. "1st tier" universities often hire the most productive faculty and have the highest expectations for promotion and tenure. Faculty at "1st tier" schools may have the best research abilities, skills, or ideas and share these ideas or skills with their students resulting in their students also becoming more successful researchers.
3. "1st tier" universities can pick the most promising new graduate students, which includes students who had the best combination of ability, skills, and motivation in undergrad and going into grad school. These students are likely to continue to perform at the top.

To answer your question directly, I believe that a student at a "2nd tier" school could certainly distinguish themselves sufficiently to be hired at a "1st tier" school. It happens infrequently because the students at "1st tier" schools have the advantages of being advised by top researchers and being top students when entering grad school.

Nevertheless, distinguishing yourself for a faculty job is hard at any school. I know many PhD alumni of "1st tier" schools who left academia.

Finally, not every school can be "1st tier", and professors at "2nd tier" schools can still do good work in research and teaching. You may want to consider whether you prefer being a professor at a "2nd tier" school to an industry job.

This is very field specific. In my field, neuroscience, where you do your PhD is probably less important that where you do your postdoc. So, if you do good work for someone who is well respected at a "2nd" or "3rd" tier school, your PhD advisor can probably help you get a postdoc at a 1st tier school. If you are then productive in your postdoc, you may have a chance at getting a faculty position.

Yes, it is possible, because the "tier" of your grad school is not something people care about anyway:

Nobody in theoretical computer science cares where you got your degree. Really. We. Do. Not. Care. We only care about the quality and visibility of your results. Publish strong papers and give brilliant talks at top conferences. Convince well-known active researchers to write letters raving about your work. Make a good product and get superstars to sell it for you. Do all that, and we'll definitely want to hire you, no matter where you got your degree. On the other hand, without a strong and visible research record, independent from your advisor, you are much less likely to get a good academic job, no matter where you got your degree.

Caveats apply, also given by JeffE in that answer:

And. In my experience, where you get your degree is strongly correlated with successful research. I got my Master's degree at UC Irvine in 1992 and my PhD at UC Berkeley in 1996. The biggest difference I saw between the two departments was the graduate-student research culture. Every theory student at Berkeley regularly produced good results and published them at top conferences. When the FOCS deadline rolled around each year, the question I heard in the hallways from other students was not "You know the deadline is coming up?" or "Are you submitting anything?" but "What are you submitting?", because "nothing" was the least likely answer. Everyone simply assumed that if you were there, you were ready and able to do publishable research. Publishing a paper wasn't exceptional, it was just what you did. That cloud of free-floating confidence/arrogance had a huge impact on my own development as a researcher. I've seen similar research cultures at a few other top CS departments, especially MIT, Stanford, and CMU. (Caveat: This is an incomplete list, and there are many departments that I've never visited.)

In this sense, universities "hire up" because graduates from top schools are much more likely to be great researchers in the first place.