Hi Michał,
Are you asking how to download all certificates that have been logged to a particular CT Log?
If so, the way to do this would be to write some code to retrieve entries from the Log in batches.
The algorithm would be similar to that defined in the CT RFC for building a Monitor:
Whether you choose to verify the signatures would be up to you.
Essentially you need to:
* get the log's current tree_size by retrieving its STH via GET <log_url>/ct/v1/get-sth
* iterate from 0 to tree_size to retrieve the log entries in batches via GET <log_url>/ct/v1/get-entries?start=<batch_start>&end=<batch_start + batch_size>
There's definitely Python client code that does the above in the certificate-transparency Github repo. I think the C++ client code you might need is there too.
hth
Paul