The text you've quoted is the D&D 3.5 version of Produce Flame, not the D&D 5e version. In D&D 5e, produce flame does not grant a melee touch attack, but instead allows the caster to throw the flames as a ranged spell attack.
Even if it did involve a melee attack it would be granting a new kind of attack option, not a bonus to damage of an existing attack, and would therefore not be usable with a monk's unarmed attacks.
Globe of Invulnerability states:
Any spell ... can't affect creatures within it, ... Such a spell can target creatures and objects within the barrier, but the spell has no effect on them. Similarly, the area within the barrier is excluded from the areas affected by such spells.
First the globe only affects spells; effects that are not spells can get through even if they are magical in nature. For example, a bard's Cutting Words, a Cleric's Turn Undead, a Monk's Ki, a Paladin's Smite, a Sorcerer's Wild Surge (unless it says "you cast ...") or a Warlock's Hurl through Hell are all unaffected. Similarly, the effects of magical items (unless they say "you cast ...") ignore the globe. (+1 spear works, magic missiles cast by wands would not).
Further, there is nothing in the wording of the spell that prevents the people and objects within the globe from being affected by spells indirectly. For example, a person standing on a rope bridge protected by the globe is still going to fall if I target the parts of the bridge outside the globe with a disintegrate spell. Similarly, I can Conjure Animals outside the globe and send them into the globe to bite you.
Then there are the edge cases.
For example, Conjure Barrage says "You throw a nonmagical weapon ... into the air to create a cone of identical weapons that shoot forward ..."
This one could be interpreted either way: if it is an area of effect type spell then the globe will block it if it is the creation of arrows, say, outside the globe which get flung by the spell then it should get through. You will need to work these ones out with your DM.
Best Answer
If you wanted to go by spell level, along the same vein as the darkness/daylight spells, then gust of wind wouldn't be able to put out a wall of flame unless it is cast as a fourth level spell.
From page 227, PHB:
Using the listed examples as the rule, Wall of Fire would be immune to being extinguished by Gust of Wind, as it is a much larger conflagration than a mere candle or torch.
From page 263, PHB:
A flame wall that is dense enough to be opaque is large and strong enough that a gust of wind won't be able to put it out. Imagine a raging inferno the size of a house, and imagine trying to put it out with air. You'd need hurricane force winds to extinguish the flames, especially since pouring air onto a fire just feeds it oxygen faster.
I don't think the source of the flame matters as much as its size. Produce flame would work well for lighting a torch, or setting easily combustible material alight (like a bale of hay), but wouldn't really stand up against a hurricane. Any fire can be extinguished when the combustion material is consumed or its access to oxygen is blocked. Wind is great for putting out smaller flames, but would do poorly against larger ones because it can't effectively smother the entire fire at once (a very poor model for describing the dynamics of fire, but it is approximately accurate). Water would be a much better choice for taking out a fire, magical or not because it can smother the entire flame, blocking the flame from accessing the combustion material (probably wood or coal) and blocking access to oxygen.