Is it obligatory to meticulously investigate the lawfulness of food by inquiring about the type of oil and the source of ingredients, and is it necessary to cleanse the mouth of food residue or plaque after it has been defiled by unlawful food or blood?

Piety (wara') and seeking out the lawful (halal) are commendable, but they should not reach the point of extremism (ghuluw) or obsessive doubt (waswas). It is not required to scrutinize and investigate the lawfulness and purity of food unless there is a reason for suspicion, for the basic principle regarding things is that they are lawful and pure. If you do not know the origin of the oil (vegetable/animal, lawful/unlawful, pure/impure), then use it without hesitation, unless you strongly suspect it to be impure, in which case you should then investigate its true nature. If the mouth becomes impure with blood or impure food, it must be purified, and any remnants of the impure food must be removed.