Publicly available data sometime still carries restrictions. These may be presented in a "click through" agreement that you agree to (don't we all) as you breeze by. This is important for data that may be still unpublished/under analysis. A common restrictions is that you can't use that data for an independent publication (especially if it is still unpublished).

In short, you would want to do due diligence, before assuming an accessible dataset is public and running with it.

If in doubt, contact the owners of the data.

I guess it is. Of course explain in your methods of your paper how you got your data and what you did with it.

It is not common for data owners to prohibit publications based on their datasets, because they also benefit from published works that reference them. What they do not want is someone scraping their site and republishing the same database content without adding much work or value.

Most often, you cannot republish the reads, but typically should be possible to publish various computational results derived from these reads by non-trivial calculations, especially such that the original data could not be recomputed from the published processed data (maybe extracting sequences from .sra files and re-zipping as .gz is not enough).

You surely need to check the licenses, but it is highly unlikely that nothing can be published, even if it is a commercial database with paywalled content.