First of all we need to find a consensus about the definition of naby and rasul.
According this (Arabic) Fatwa -my translation from Arabic, please take it with care- the term naby is more general than the term rasul: some scholars define naby as somebody who has received a revelation or message from Allah if he has been asked to spread it then he will be called rasul and naby if not only naby. Others said that a rasul is somebody who has received a revelation with a new shari'a (which he has to spread), while a naby has been sent out to confirm a shari'a revealed to those who came before him. According the fatwa the first opinion is stronger.
Note that logically the last statement means if Adam (peace be upon him) had a shari'a he necessary need to be a naby and rasul. On the other hand there's no statement in the quran quoting Adam (peace be upon him) clearly as a rasul, while there are many statements which may indicate that he at least was a naby, like his speech to Allah.
- Sheikh ibn 'Utaymyn only confirmed Adam (Peace be upon him) to be a naby but not a rasul based on Verse (2:213):
Mankind was [of] one religion [before their deviation]; then Allah sent the prophets as bringers of good tidings and warners
According this Arabic fatwa and to the Saudi Arabian fatwa council (7701) .
- While this fatwa says Adam (peace be upon him) -my translation- was a naby but he came with a shari'a which he taught his children, which means he is also a rasul to his children (according the above definition) they also quote a hadith from Musnad al-Imam Ahmad on the authority of abu Dharr (which quotes that Adam was a naby, who had spoken to Allah):
عن أبي ذر رضي الله عنه قال: دخلت على رسول الله صلى الله عليه وسلم وهو في المسجد: فقلت: أي الأنبياء أول؟ قال: آدم، قلت: وهل كان نبياً؟ قال: نعم، نبي مكلَّم
The mufassirin said that a part of the shari'a of Adam, was that a man couldn't marry his "twin" sister (more exactly a sister with whom his mother was pregnant beside him) ... which by conclusion makes Adam a naby and rasul.
This fatwa (Arabic) adds some more references to support Adam having talked to Allah (2:33) and for his shari'a (5:27-28). Also a hadith quote from the book a-dalaail of Imam al-Bayhaqi has been added.
Again according this fatwa (Arabic) Adam (peace be upon him) is the first naby and rasul again adds as to support the claims (2:31)
And He taught Adam the names - all of them. ...
and the story of Qabeel and Habeel from surat al-Maaida (5). And adds that one problem is the hadith about shafa'a on the day of resurrection (here the Version from sahih al-Bukhari), where Adam is only called the father of mankind while calling Nuh (Noah) the first of Allah's rusul/messengers. But according to a comment (tuhfat al-ahudi) of sunan at-Tirmidhi of this hadith quoting al-qadi 'Iyad Adam and Sheet (peace be upon them) were both rusul/messengers before Nuh. Again my translation take it with care-... while Adam has been sent to his children (who have not been disbelievers/kuffar) to teach them iman and obeying Allah. And Sheet (Seth) has followed him later while Nuh was sent to the disbelievers on earth and this is closer than saying Adam and Idriss weren't rusul/messengers!
قال القاضي عياض : ... وأما آدم وشيث فهما وإن كانا رسولين إلا أن آدم أرسل إلى بنيه ولم يكونوا كفارا بل أمر بتعليمهم الإيمان وطاعة الله . وشيث كان خلفا له فيهم بعده بخلاف نوح فإنه مرسل إلى كفار أهل الأرض وهذا أقرب من القول بأن آدم وإدريس لم يكونا رسولين
Note that qadi 'Iyad, in this quote, was also speaking about 'Idriss whom he has also named Ilyas.
A short comment on verse 3:33
I couldn't find neither in tafsir ibn Kathir, nor in tafsir at-Tabari and tafsir al-Baghawi, nor in tafsir al-Qurtobi, al-bahr al-moheet or at-tahriri wa-t-tanwir any indication about this verse being used to support the claim that Adam (peace be upon him) is a rasul. All of them agree that he has been chosen by many things: he is the father of mankind, Allah Himself has made Adam and giving him a soul and giving him his name and he has taught him the names of everything, He has settled him in Janah for a while and settled him down on earth later for a reason only He knows. Most of the tafsirs indicate that this verse is an answer to Jews (some say Christians or both) who claimed to be the chosen ones or follow the religion of the chosen ones...
What we can definitely say is that Adam (peace be upon him) was a naby with his own shari'a. Assuming that a rasul needs to spread the message while a naby is not asked to or a rasul is a person who has a new message while a naby only confirms an old one logically Adam should be considered as a naby and rasul.
There are no quotes from quran clearly saying neither that Adam is a naby nor a rasul, but one could easily conclude the first. In sunnah the strongest evidences say that Noah (peace be upon him) was the first rasul, but some scholars explained that as follows: at the time of Adam there were no disbelievers that's why he isn't called rasul even if he had a message and a "new" shari'a.
And Allah knows best!