What are the benefits of designing paired vs unpaired clinical experiments I would like to know what additional information one can obtain from paired vs unpaired clinical data? Why one should prefer one design over the other one?
My second question, imagine I run two separate trials, testing a drug, one being a paired design and the other being unpaired. If I run the paired ttest on the first trial data and unpaired ttest on the second dataset, would there any connection between the two tests outcomes? If I have the paired design run first, can I use the outcomes as expectation/priors for the unpaired data?

In general, the paired design will provide more information.  The 'before' measurement should provide an unbiased estimate of what the control arm would be, whereas the 'after' measurement should provide an unbiased estimate of what the treated arm will be.  In addition, you can get an estimate of the before-after correlation, which is unavailable from the unpaired design.  Given that the before-after correlation is typically positive (and at least moderately strong), the paired design will afford greater statistical power.
