I know that the Persian language family has three branches: Persian, Dari and Tajik-Persian.
Is the Persian of Iran mutually intelligible with Tajik?
And how does the Tajik Cyrillic alphabet differ from the Russian Cyrillic alphabet?
Yes, the standard forms of the Persian of Iran, Afghan Persian, and Tajiki are mutually understandable. They are about as different as British and American English. But this does not mean that all dialects are likely to be understood in the other countries, or even in different regions of the same country.
Yes, modern standard Tajik, Dari and Iranian Persian are mutually intelligible.
All three descend from Middle Persian. In some cases the Tajiki variant is actually truer to older or more formal Persian.
The differences in language names and alphabets are largely political, or social, not linguistic, and has changed over time.
The variants do differ in phonology, grammar and vocabulary, especially in informal speech. Notably, each name covers dialects that also differ significantly. (For example, Tajik in China is better grouped among Pamiri languages.)
The Tajik Cyrillic alphabet is still the current standard de facto alphabet in Tajikistan and other post-Soviet Republics where Tajik Persian is used.
There are now 5 letters not found in the Russian Cyrillic alphabet.
There are also a few letters found in Russian Cyrillic not used in Tajik Cyrillic.
Necessarily, the precise pronunciation of all the letters differs from Russian.
In the past there were other versions of the Tajik Cyrillic alphabet, and Tajik has also been written in Latin, Perso-Arabic and Hebrew script.