Do muslims find Musnad Ahmad and Tafsir ibn Jarir at-Tabari as authentic sources of hadith? I have heard that there are many Daeif hadiths in these sources. Also, do Muslims believe in only the Sahih hadiths to be fully authentic or do Muslims believe in both Sahih and Hasan hadiths?
*JamI' al-Bayan 'an ta'wil ayi al-Qur'an جامع البيان عن تأويل آي القرآن: Tafsir at-Tabari**
The tafssir of imam ibn Jarir at-Tabari is among the first Tafssir books and one of the best examples of at-Tafssir bil Ma'athur (see for example Are al-Tabari's Commentary (10th century) & Ibn Kathir's Commentary (14th century) on the Qur'an available online?). This is a tafssir which largely relies on narrations with a sanad -narrator chain- (mostly full and connected sanad) which goes back to the source of the information be it the prophet () himself, a sahabi() or a tabi'i etc. . Beside this greate book you may also find among the oldest Tafassir that of ibn abi Hathim who is also known as a greate hadith scholar with qualities that are comparable to those of imam al-Bukhari or Muslim or an-Nasa-i etc..
In fact it doesn't make much sense to consider a Tafssir book as a source of authentic hadith as it contains narrations which don't stricly speaking are considered hadith and we know for sure that the Tafssirs of at-Tabari and ibn Kathir contain some so called Israeliyat which are more or less hair rising stories that have been spread by the people of the book. Note that even if at-Tabari came after al-Bukhari and Muslim he did efforts to quote ahadith with his own narrator chains which may be covered in the sahih books.
He was a great scholar and even founded his own madhhab, but he was fought by the followers of the hanbali school of fiqh in Baghdad who claimed that he was a Shi'a sympathisier. He actuall had his own hadith collection called Tahdheb al-Athaar هذيب الآثار ad-Dhahabi says that he died before completing it.
As for Musnad Ahmad I think that I have explained and covered the matter with enough details in my answer on Is Musnad Ahmed a book of Sahih Ahaadith?.
Just as a summary and addition: imam Ahmad in his Musnad was not intending to compile only sahih hadith, but he compiled all kinds of hadith which are used as evidences by scholars so it covers all the sahih narrations of his student al-Bukhari and Muslim etc., as well as hassan narrations, weak narrations and also a couple of fabricated narrations.
Also be aware that imam al-Bukhari in his sahih explicitly says that his quotes are sahih narrations only when they have a full isnad (full narrator chain), so not all narrations in the book fall under this category.
When is a hadith relaible? (Sunni view)
Finally a hadith is reliable when it is considered as sahih or at least hassan. But in some cases -when it comes to good behaviour, manners, dhikr etc.- it is allowed to apply da'if narrations. Note that this is a general statement based on the view of hadith scholars, when it comes to jurisprudence details of each school must be considered.
On the authentication of hadith: at-Tabari vs. Ahmad
If we try a comparison between imam ibn Jarir at-Tabari and imam Ahmad when it comes to accepting the narrations of certain narrators we can safely say that ibn Jarir at-Tabari is more lenient than imam Ahmad as he accepted a whole bunch of "unknown" narrators in cases even knowing that others rejected them without explaining why he considered their narration as reliable.
Sahih Bukhari and Sahih Muslim are what we taught as Muslims to believe and trust. They are reliable sources of Hadith, not because we have been taught that in schools and Mosques, but because they have strong Sanad in every Hadith that is stated in their books.
As for Musnad Ahmad and Tafsir ibn Jarir at-Tabari, if they contain weak hadiths, then you better skip them. If you encounter a Hadith that you are unsure of its reliability, then the standard procedure is to look at its Sanad and follow the narrators in the ascending order of the year of their death. If a conflict happens regarding their order in the Sanad of that Hadith, then it's considered weak, and should be skipped.
A first step that you may consider doing before checking the Sanad is to find if there's any contradiction within the Hadith saying and the Qur'an. Since what the prophet (PBUH) says in Hadiths is impossible to conflict with what Allah says in Qur'an.
For more details, check this great answer in Islam Exchange