1 question

Question

If I need a book/sefer for school, can i go to a Jewish Library in a Conservative Shul? The shul - if you want to call it that - has a very large library that may have sefarim i need for a Jewish Law paper I would like to write. I do not think I will be entering the actual Shul at all? 2) in my city there is a bagel shop that is owned by a non-shomer shabbos jew. This bagel shop a long time ago used to have a hechsher, however, it does not have one nowadays, primarily because it is open on shabbos and does not have a mashgiach. The grandson of the owner is Frum and told me that the Bagels were and still are 100% kosher. If he watches one of the bagel 'runs' and gives me bagels from that run and tells me they are OK, can I trust him??? Thank you