Bane has no bearing on a Critical Hit.
From page 194 of the PHB:
If the d20 roll for an attack is a 20, the attack hits regardless of any modifiers or the target’s AC. In addition, the attack is a critical hit, as explained later in this chapter.
Since a Critical Hit is determined by the roll of a natural 20, and not by the end result adding up to 20, any addition or subtraction (such as Ability Modifiers [as stated above] or spells such as Bane) to this roll doesn't matter, it is still a Critical Hit.
Also note that even if that were not the case, any creature that has +4 or higher bonus to the attack roll essentially counteracts the effects of Bane anyway.
No, Bardic Inspiration can't cause a critical hit
Cutting Words states, "...subtract the number rolled from the creature's roll.", so the number is being applied to actually change what you rolled on the d20.
Bardic Inspiration states, "add the number rolled to one ability check, attack roll, or saving throw", so the number is just another bonus on the attack roll.
The difference in phrasing is very small but important here. Terms like, "attack roll", "saving throw", "ability check", are common phrases in 5e used throughout the book. PHB, p.195, explains that you make an attack roll by rolling a d20 and adding appropriate modifiers. If it weren't for the Jeremy Crawford tweet it would be sensible to assume that the, "roll", in Cutting Words refers to the, "attack roll", but that tweet exists so instead it refers to the actual roll of the d20. Bardic Inspiration specifically says, "attack roll", so we can be relatively certain that it is just another modifier added to the roll.
The spell Bless uses the same phrasing as Bardic Inspiration, "...and add the number rolled to the attack roll or saving throw", and there is no reason to believe that it grants (or bizarrely prevents) critical hits.
Best Answer
Temple of the Gods modifies the d20 roll, and modifiers don't cancel critical hits
The steps for making an attack are described as follows (emphasis added):
Temple of the Gods says to "subtract the number rolled from the d20 roll"; Bane says to "subtract the number rolled from the attack roll or saving throw"; and Cutting Words says "subtracting the number rolled from the creature's roll". Regardless of the minor variations in wording, all of these effects, including Temple of the Gods, are describing penalties or bonuses to the d20 roll, which are covered above in the bold-face text in step 2. Notably, these modifiers are all determined before the d20 roll, which happens in step 3. This means that, somewhat counterintuitively, the technically correct order of operations is to roll the d4 first, incorporate it into the attack modifier, and then roll the d20 and add the modifier you determined in step 2. (Obviously, the order in which the dice are rolled doesn't actually affect the result, and I don't know anyone who rolls this way in practice.)
As the answers to the linked questions have adequately covered, modifiers can't turn a critical hit into a regular hit. The relevant phrase from the rules is (in bold):