No, Two-Weapon Fighting does not apply.
Magic stone is neither a weapon, nor melee, nor light.
Two weapon fighting requires that you be using a light melee weapon. Melee weapons can be used to make melee weapon attacks.
The magic stone spell creates objects with which you can make a ranged spell attack only. This spell attack uses your spellcasting modifiers for attack and damage, regardless of who is using the stone. This spell attack does not add to the pebbles' normal damage (like magic weapon or hex or smite) but replaces it completely. This indicates that the attack used in magic stone is not based on any weapon-like properties, so you would be at a stretch to classify the magic stones as weapons. If they are not weapons, they cannot qualify for two weapon fighting.
There is still a bit of grey area, though. In principle a regular pebble can be used as an improvised weapon, so it is possible to argue that magic stones are weapons too. However, because magic stones can make only ranged attacks and not melee attacks, they are at best ranged weapons, not melee weapons. (The dart and net are ranged weapons which also have the thrown property. This means that being thrown does not imply being a melee weapon.)
Because the magic stones are (at best) ranged weapons and not melee weapons, they cannot be used with two weapon fighting, even if you could argue that they were light.
But I shall also argue why magic stones (and pebbles generically) are probably not light weapons either. From PHB 147,
A light weapon is small and easy to handle, making it ideal for use when fighting with two weapons.
The definition of a light weapon makes no reference to a weapon's weight. A critical part of being 'light' is being "easy to handle", such that it is ideal for use when two-weapon fighting. Rapiers and warhammers are 2 lb. and are not 'light'. A dart, which is the closest listed weapon to a thrown stone, is only 1/4 lb. and is also not 'light'. Because a dart is not 'light', I see no reason to assume that a magic stone is 'light' either.
You are asking this question because you wish to get more damage at a range. Consider that ranged combat already comes with significant advantages, namely you get to stay out of reach of people with melee attacks. The higher damage potential (from two weapon fighting, opportunity attacks, and availability of stronger weapons) is the primary benefit of melee. If you remove this advantage from melee, the reasons for using melee become fewer and balance suffers. Choosing ranged over melee is a trade-off.
If you want to optimise your damage output from magic stone, get two or three friends (or hirelings or familiars or conjured creatures) to help you throw the stones. By using allies, you can get three attacks every turn from magic stone, which is twice as fast as your proposed scheme of three attacks over two turns.
Best Answer
Yes, you can.
Compare that phrasing to the third benefit of the War Caster feat (PHB, p. 170):
You can see that it specifies what type of cantrip you can cast, with the limitations of needing to have a normal casting time of 1 action, and also only targeting the creature that provoked the opportunity attack. It effectively limits your options to mostly just offensive cantrips.
Notably, the Bladesinger's Extra Attack feature lacks any language that restricts what type of cantrip you can cast. It simply states that "you can cast one of your cantrips". Obviously, the intent was for you to cast an attack cantrip, such as booming blade or fire bolt, but this ability has been worded poorly, and allows for any cantrip to be cast, including ones that normally have a casting time of "1 bonus action" or "1 minute".
In the case of mending, it should go into effect immediately, although the benefits of casting mending in the middle of combat are few, unless you have an Artificer ally with an Eldritch Cannon, Steel Defender, or Homunculus Servant in need of repair.
If I were the DM in this situation, I would restrict it to only cantrips with a casting time of 1 action, and disallow things like the examples you provided. I would still allow non-offensive cantrips, as I like to reward creative uses of spells.