1. You are both an Arcane and a Divine caster
By any sane definition, being able to cast Cleric spells as Divine spells makes you a Divine caster. You already were an Arcane caster.
And yes, this qualifies a pure Wizard X/Rainbow Servant 10 (given your ability to cast a 2nd level Divine spell) for Mystic Theurge. Though...
2. You only gain one spellcasting level
The full text of the Spells per Day feature of the Mystic Theurge prestige class:
Spells per Day: When a new mystic theurge level is gained, the character gains new spells per day as if he had also gained a level in any one arcane spellcasting class he belonged to before he added the prestige class and any one divine spellcasting class he belonged to previously. He does not, however, gain any other benefit a character of that class would have gained.
This essentially means that he adds the level of mystic theurge to the level of whatever other arcane spellcasting class and divine spellcasting class the character has, then determines spells per day and caster level accordingly.
If a character had more than one arcane spellcasting class or more than one divine spellcasting class before he became a mystic theurge, he must decide to which class he adds each level of mystic theurge for the purpose of determining spells per day.
Note the bolded part in the second paragraph, which you did not quote in the question. The class feature Spells per Day of the Mystic Theurge finishes off by explicitly telling you how the effect works - you add your level of Mystic Theurge to the level of whatever other Arcane or Divine class you have, though (as per the third paragraph) only one of each.
Note the key phrase, 'adds the level of mystic theurge to the level of whatever other [...] spellcasting class the character has'.
Base spellcasting class levels
+ Rainbow Servant spellcasting progression levels (6/10 or 10/10, YMMV)
+ Mystic Theurge levels
= Total spellcasting level
Adding Mystic Theurge again to the total goes against only adding Mystic Theurge to "other" spellcasting classes, as Mystic Theurge is already included.
3. Caster level is not spellcasting levels
Your phrasing makes it clear that you are asking for the latter, "if a character's effective levels in a casting class exceeds 20, does the slot progression continue?" However, as you asked about "caster level", I need to specify that Caster Level is a specific term used to determine the effectiveness of your spells.
For instance, the Master Specialist prestige class (Complete Mage, p.70) includes the class feature "Caster Level Increase" which is defined as such:
Caster Level Increase (Ex): Upon reaching 6th level, add 1 to your caster level whenever you cast a spell of your chosen school. At 9th level, you instead add 2 to your caster level.
As to your actual question, yes. Unless your DM rules that Epic content is disallowed, if your character's effective level in a spellcasting class exceeds the non-epic scale, so would the character's spellcasting ability. However, a look at the Epic Class Progression for Wizard leaves us with:
Spells: The wizard’s caster level is equal to her class level. The wizard’s number of spells per day does not increase after 20th level. Each time the wizard achieves a new level, he or she learns two new spells of any spell levels that he or she can cast (according to his or her new level).
...which is a bit anticlimactic. You would only gain further spells known, not even access to Epic Feats. The same holds true for the other (PHB) base casting classes. Probably time for some levels of Wonderworker (BoED, p.82) and extra spell slots instead.
My thoughts and rulings (not RAW)
If your Wizard X/Rainbow Servant 10 is both an Arcane caster and a Divine caster, does that mean the character now has a spellcasting Class that is both Arcane and Divine? No, not in my opinion. The character's spellcasting Class is still Wizard, which is an Arcane spellcasting Class.
A Sha'ir would be a more troublesome example, where the class itself already casts both Divine and Arcane spells. Here I would have to resort to the fluff, and read the base description of the class itself:
Between commerce, diplomacy, or even war, the sha'ir's arcane traditions could slowly filter across the world.
as well as another class feature:
Weapon and Armor Proficiency: Sha'irs are
proficient with all simple weapons, but not with any
type of armor or shield. Armor of any type interferes
with a sha'ir's arcane gestures, which can cause his
spells with somatic components to fail.
Not nearly as conclusive as Wizard X/Rainbow Servant 10, IMO, but good enough for me to make a ruling at my table. The Sha'ir, much like a Wizard X/Rainbow Servant 10, is (by my house rules) an Arcane spellcasting class with access to Divine spells via Domains, which are cast as Divine spells.
A Domain Wizard with the Storm Domain, on the other hand, explicitly casts certain Cleric and Druid spells as Arcane spells, which is "merely" an expansion of the Wizard's spell list.