Here is a quickly go over only.
I. Taking the words of the seller concerning components and process
SA YD siman 119 adress the subject.
There are several cases.
The basic question is if you can believe that food is Kasher when you ask someone who himself eats non Kosher foods (sayif 1), who doesn't eat but sells to others non Kasher foods (sayif 2) ; if someone who is unreliable for a rule X can be asked about a rule Y (sayif 4) ; if you provide ingredients, can we have confidence that the dispenser uses them only (sayif 3).
Concerning non Shomer Shabbat Jew see sayif 7, it seems that who is Mechallel Shabbat is unreliable for Kashrut questions.
If you ask about buying Kosher food in a Jewish store which sells non Kosher food or is known as lying regarding some products origin, there is a knass (sayif 4) to not buy from him even foods which are doubtless.
Are there any exceptions?
To buy Kosher food in a non Kosher store.
To buy Foods which potential prohibition Deorayta, as foods containing wine, meat, fish, we need to see a double sealed (שתי חותמות). Potential prohibition Derabanan, needs one sealed only. Some poskim require double sealed for Deorayta problems with suspected Jews only, with non Jew, one sealed is good enough. Bread which can be manufactured by a non Jew owner needs to be sealed (because the maximum of leniency concerning Pat Akum is for pat palter, not the bread which the NJ made at home for himself.) (SA YD 118, 1).
II. Non Jew's made foods
The prohibition of Bishul Nochri.
mainly treated in SA YD 113.
Let's adress the problem of bishul (when there is no problem with ingredients).
The prohibition of Bishul Nochri apply when two conditions are fulfilled: 1. not eaten raw, 2. brought at kings table. (Or a mixture of foods in which the main part meets this conditions). Hard boiled eggs are prohibited.
Roasted Chickpeas ( ארבעס) and similar seeds are allowed, small fried fishs.
Bakery bread is a sufficiently known issue. (according to some poskim if the bread is brush with eggs yolk, there is no kula of pat palter, because of the eggs)
Could I eat kosher food cooked by my mother if, G-d forbid, she weren't shomer Shabbos?
The question is if the prohibition of Pat Kutim shel baale Batim is extended to Jewish people who has a status of Mumar (renegade). Concerning non Jew's products, wine and bread have similarities, "Stam Yenam" and "Pat Nochri", They result from decrees because of inter marriage risk, poskim (SA YD 112, 1 Baer Heytev sk 1) discussed Hava Amina to allow bread of monks (who have no children). For a renegade, in contrast with wine, there is no problem of chatnut Lechaora and the pat is allowed. It seems that the Gezera for Mumar is limited to wine only (SA YD 124, 8)