Is it mustahabb to offer du‘aa’ when rain is falling? What should be said when rain falls and when hearing thunder?

Praise
be to Allah

Firstly:

It was narrated from ‘Aa’ishah (may Allah be pleased with
her) that when the Messenger of Allah (blessings and peace of Allah be upon
him) saw rain, he would say: “Allahumma sayyiban naafi‘an {O Allah,
(make it) a beneficial downpour).” Narrated by al-Bukhaari (1032)

According to a version narrated by Abu Dawood (5099), he used
to say: “Allahumma sayyiban hanee’an (O Allah, (make it) a wholesome
downpour).” Classed as saheeh by al-Albaani.

The word sayyib (translated here as downpour) refers to rain
that flows. Allah, may He be exalted, says (interpretation of the meaning):
“Or
like a heavy downpour [sayyib] from the sky”
[al-Baqarah 2:19].

See: Ma‘aalim as-Sunan by al-Khattaabi (4/146)

It is mustahabb to expose oneself to the rain and let some of
it flow over one’s body, because of the proven report from Anas (may Allah
be pleased with him) who said: When we were with the Messenger of Allah
(blessings and peace of Allah be upon him) it rained. The Messenger of Allah
(blessings and peace of Allah be upon him) lifted part of his garment so
that the rain could fall on him. We said: O Messenger of Allah, why did you
do that? He said: “Because it has just come from its Lord, may He be
glorified and exalted.” Narrated by Muslim (898).

When the rain grew severe, the Prophet (blessings and peace
of Allah be upon him) said: “Allahumma hawaalayna wa la ‘alayna;
Allahumma ‘ala al-aakaami wa’z-ziraabi, wa butoon al-awdiyah wa manaabit
ash-shajar (O Allah, (let the rain fall) around us and not upon us, O
Allah, (let it fall) the small mountains and hillocks, the valley bottoms
and places where trees grow).” Narrated by al-Bukhaari (1014).

With regard to du‘aa’ when hearing thunder, it is proven from
‘Abdullah ibn az-Zubayr (may Allah be pleased with him) that when he heard
thunder he would stop speaking and would say: Subhaan alladhi yusabbih
ar-ra‘du bi hamdihi wa’l-malaa’ikatu min kheefatihi (Glory be to the One
whom the thunder glorifies and praises, and so do the angels because of His
Awe) [cf. ar-Ra‘d 13:13]. Then he would say: This is a stern warning to the
people of earth. Narrated by al-Bukhaari in al-Adab al-Mufrad (723);
Maalik in al-Muwatta’ (3641). Its isnaad was classed as saheeh by an-Nawawi
in al-Adhkaar (235) and by al-Albaani in Saheeh al-Adab al-Mufrad
(556).

We do not know of any marfoo‘ report from the Prophet
(blessings and peace of Allah be upon him) concerning that.

Similarly, there is no proven dhikr or du‘aa’ from the
Prophet (blessings and peace of Allah be upon him) for seeing lightning, as
far as we know. And Allah knows best.

Secondly:

The time when rain comes down is a time of divine bounty and
mercy from Allah to His slaves, when the means of goodness are abundant, and
it is a time when it is thought that du‘aa’s will be answered.

It says in the marfoo‘ hadith of Sahl ibn Sa‘d that the
Prophet (blessings and peace of Allah be upon him) said: “Two (du‘aa’s) are
not rejected: du‘aa’ at the time of the call to prayer and du‘aa’ at the
time of rain.”

Narrated by al-Haakim in al-Mustadrak (2534); at-Tabaraani
in al-Mu‘jam al-Kabeer (5756); classed as saheeh by al-Albaani in
Saheeh al-Jaami‘ (3078).

And Allah knows best.