Could you tell me some of the deeds that the Messenger of Allah did during the Makkan period that indicate his confidence that he was indeed the Messenger of Allah?

After the first verses of Soorat al-Muddaththir were revealed
to the Messenger of Allah (blessings and peace of Allah be upon him),
instructing him to call people to Allah and convey the religion, from that
moment he started to strive persistently, based on great faith and
certainty, and his efforts to convey the message never faltered.

He began to go around to the polytheists, conveying to them
the command of Allah and telling them that He alone was deserving of
worship, to the exclusion of all others, and that he had been sent from
Allah to revive what had been forgotten of the teachings of the Messengers
before him, reciting to them the verses that had been revealed concerning
that.

Allah, may He be exalted, said, instructing His Prophet in
Soorat al-A‘raaf, which was revealed during the Makkan period
(interpretation of the meaning):

“Say (O Muhammad (blessings and
peace of Allah be upon him)): ‘O mankind! Verily, I am sent to you all as
the Messenger of Allah — to Whom belongs the dominion of the heavens and the
earth. Laa ilaaha illa Huwa (none has the right to be worshipped but He); It
is He Who gives life and causes death. So believe in Allah and His Messenger
(Muhammad (blessings and peace of Allah be upon him)), the Prophet who can
neither read nor write (i.e. Muhammad (blessings and peace of Allah be upon
him)) who believes in Allah and His Words, and follow him so that you may be
guided”

[al-A‘raaf 7:158].

How could he not be certain of that, when it was Allah, may
He be exalted, Who was instructing him to proclaim to all people that he was
the Messenger of Allah to them?

And He says in Soorat al-Haaqqah, which was also revealed
during the Makkan period (interpretation of the meaning):

“So I swear by whatsoever you
see,

And by
whatsoever you see not,

That this
is verily the word of an honoured Messenger [i.e. Jibril (Gabriel) or
Muhammad (blessings and peace of Allah be upon him) which he has brought
from Allah].

It is not
the word of a poet, little is that you believe!

Nor is it
the word of a soothsayer (or a foreteller), little is that you remember!

This is
the Revelation sent down from the Lord of the ‘Alamin (mankind, jinn and all
that exists)”

[al-Haaqqah
69:38-43].

And there are many similar verses.

Anyone who any knowledge of the religion of Islam will know
for certain that the Messenger of Allah (blessings and peace of Allah be
upon him) did not have any doubt that he was the Messenger of Allah. How
could he have any doubt about that, when he was calling people to enter
Islam by testifying that there is no god but Allah and that Muhammad is the
Messenger of Allah?

How could the Messenger of Allah (blessings and peace of
Allah be upon him) not be confident of his Prophethood when he passed by his
companions being tortured by the polytheists, until Sumayyah, the mother of
‘Ammaar ibn Yaasir, was killed, and the Prophet (blessings and peace of
Allah be upon him) said to them, “Patience, O family of Yaasir, for you are
promised paradise”? Classed as saheeh by al-Albaani in Takhreej Fiqh as-Seerah
(p. 103).

How could he not be confident of his Prophethood during the
Makkan period when he depicted the truthfulness of his Prophethood by
saying: “Do you think that if I told you that there was a cavalry emerging
from the foot of this mountain, would you believe me?” They said: We have
never known you to be a liar. He said: “I am a warner to you of an imminent
and severe punishment.” Abu Lahab said: May you perish! Did you call us
together only for this? Then he stood up, and this soorah was revealed:
“Perish the two hands of Abu Lahab (an uncle of the Prophet) and perish he!”
[al-Masad 111:1].

Narrated by al-Bukhaari (1394) and Muslim (208).

How could he not be confident of his Prophethood, when he
answered his uncle Abu Taalib, when he said to him, O son of my brother,
your people have come to me saying such and such – describing what they said
to him – so spare me and yourself, and do not put on me a burden greater
than I can bear. The Messenger (blessings and peace of Allah be upon him)
thought that his uncle was going to forsake him and stop protecting him, and
that he was no longer able to support him and stand by him. He replied: “O
my uncle, by Allah, if they put the sun in my right hand and the moon in my
left on condition that I abandon this cause, before Allah has made it
victorious, or I perish therein, I would not abandon it.”

Narrated by Ibn Ishaaq in al-Maghaazi (1/154)

Al-Albaani said in as-Silsilah ad-Da‘eefah (2/311):

I found another chain of narration of the hadith with a hasan
isnaad, but it says: “I am not more able to give that up for you than you
are able to get me a firebrand from the sun.” I have narrated it in al-Ahaadeeth
as-Saheehah, no. 92. End quote.

How could he not be confident of his Prophethood, when he was
speaking of his ascent to heaven, his meeting with the Prophets, his coming
close to the Lord of Glory, and what He enjoined upon him of the prayers,
and the other details in the story of the Isra’ and Mi‘raaj (Night Journey
and Ascent to Heaven)? Would he tell of all of that when he was still
uncertain about his Prophethood and that Allah had sent him?

How could he not be confident of his Prophethood, when he was
reciting to the polytheists verses that spoke of his Prophethood and refuted
their doubts concerning it, when they stubbornly refused to believe that he,
of all of them, would be the Messenger of Allah?

“And they say: ‘Why does this
Messenger (Muhammad (blessings and peace of Allah be upon him)) eat food,
and walk about in the markets (as we). Why is not an angel sent down to him
to be a warner with him?

‘Or (why)
has not a treasure been granted to him, or why has he not a garden whereof
he may eat?’ And the Zaalimoon (polytheists and wrong-doers) say: ‘You
follow none but a man bewitched.’

See how
they coin similitudes for you, so they have gone astray, and they cannot
find a (Right) Path”

[al-Furqaan 25:7-9].

How could he not be confident of his Prophethood, when he was
refusing to compromise with the polytheists and rejecting what they offered
to him of  kingship, power, wealth, status and everything that if it were
offered to someone who was not confident, he would accept it?

Quraysh sent ‘Utbah ibn Rabee‘ah, who said to the Messenger
of Allah (blessings and peace of Allah be upon him):  O son of my brother,
you are, as you know, of noble descent and you have brought your people a
grievous matter because of which you have divided their community. Listen to
me; I shall give you some options, so that you may choose one of them. If
you want wealth, we will collect money from our wealth so that you may be
the wealthiest of us. If you want honour, we will make you our leader, and
we will not decide any matter without you. If you want sovereignty, we will
make you our king. If what comes to you is something caused by the jinn, and
you think that you cannot rid yourself of it, then ask us for medicine, and
we will spend our wealth on it until you are healed.

When he had finished speaking, the Messenger of Allah
(blessings and peace of Allah be upon him) recited to him the beginning of
Soorat Fussilat, up to the verse (interpretation of the meaning):

“But if they turn away, then say
(O Muhammad (blessings and peace of Allah be upon him)): ‘I have warned you
of a Saa‘iqah (a destructive awful cry, torment, hit, a thunderbolt) like
the Saa‘iqah which overtook ‘Aad and Thamood (people)”

[Fussilat 41:13].

Narrated by Ibn Abi Shaybah in his Musannaf (36560);
classed as saheeh by al-Albaani in Saheeh as-Seerah (p. 159).

After all of this, could he have had any doubts about his 
ownProphethood and the fact that Allah had indeed sent him?

But they kept trying to make deals with him and trying to
offer him compromises, in the hope that he would respond in kind and
compromise with them, as Allah, may He be exalted, says (interpretation of
the meaning):

“They wish that you should
compromise (in religion out of courtesy) with them, so they (too) would
compromise with you”

[al-Qalam 68:9].

But no, that never happened.

They even said to him: Worship our gods for one year, and we
will worship your God for one year. Then Allah, may He be exalted, revealed
soorat al-Kaafiroon (interpretation of the meaning):

“Say (O Muhammad (blessings and peace of Allah be upon him)
to these Mushrikoon and Kaafiroon): ‘O Al-Kaafiroon (disbelievers in Allah,
in His Oneness, in His Angels, in His Books, in His Messengers, in the Day
of Resurrection, and in Al-Qadar)!

‘I worship not that which you worship,

Nor will you worship that which I worship.

‘And I shall not worship that which you are worshipping.

‘Nor will you worship that which I worship.

‘To you be your religion, and to me my religion (Islamic
Monotheism)’”

[al-Kaafiroon 109:1-6].

That put an end to this ridiculous attempt at compromise. If
we were to quote all the signs of his confidence in his Prophet hood, we
would not have enough room.

And Allah knows best.