Actually, while every sentence is perfectly understandable (though a1, b1 and b2 don't sound correct in my ears, and a2 is incorrect word usage - although grammatically correct), I'd use none of those.
I'd either use:
Ich verbringe jeden Tag zwei Stunden damit, Deutsch zu lernen.
Ich finde jeden Tag zwei Stunden Zeit für's Deutsch lernen.
But the shortest way is usually best, so I'd actually use:
Ich lerne jeden Tag zwei Stunden Deutsch.
No need for any zu.
Usually when you use "zu", it's an indication that your sentence is too long, passive voice, or that your formulation is too complicated.
As for the difference between "zu", and "um ... zu":
The word "zu" has several meanings.
It can mean "towards", and as such is some kind of a mixture between preposition and adverb of place, although it technically is a preposition.
"Um zu" can be properly translated with "in order to".
So every time any sentence has some kind of "directional" meaning (e.g. ich gehe zu, zu Gunsten von [die Gunst "geht" zu jemandem] ), it is less likely there is an "um".
This "directional" meaning can also include "quantity" (bis zu, ab und zu) and property (gehören zu, zählen zu).
Every time this "directional" meaning is NOT present, you usually can (or have to) use "um zu".
"zu" can also have the meaning of "too" (as in too many), in which case the the "um" is never used.
Also "zu" can be an adjective (zu as in "Der Laden ist zu." - the shop is closed.), or a part of a noun, in which case "um" is never used.
Use "um" only when
- the word is used as a preposition
- the context does not imply any kind of direction/ownership
or when you'd use "in order to" in English.
Ich brauche das Geld, um nach Hause zu fahren.
I need the money in order to get home.