Wall of Force does not transfer thermal energy.
The Player's Handbook describes the spell as:
immune to all damage
Cold and fire are both types of damage, and in your example we can assume that the lava would cause fire damage.
The question, then is "Can Wall of Force allow damage to pass through it if it does not take damage itself?" The answer still is no, and here's why:
There are three types of heat transfer: Conduction (contact between two objects, such as in the spell Heat Metal), Convection (which happens primarily in gases and liquids, of which the Wall of Force is neither), and Radiation (does not require a medium, and consists of electromagnetic waves).
We can rule out Convection and Conduction, since the Player's Handbook describes Force as:
pure magical energy
which more than likely does not behave the same as normal matter. In addition, if Force did act in a similar way, then we have a precedent for an amount of Fire damage that does not pass through a Wall of Force, namely a Fireball spell cast at 9th level. This will do the base 8d6 fire damage, plus an additional 5d6 fire damage for a maximum of 78 fire damage. Given that there is no limit to the number of creatures that can take a turn in a round, we can assume that even with even three spellcasters casting 9th level Fireball spells, the wall (and the creature being protected by the wall) will remain unharmed.
Per the Improvising Damage chart on page 249 of the Dungeon Master's Guide,
18d10 Being submerged in lava, ...
That's a maximum of 180 fire damage per round submerged in lava. With 3 9th-level Fireball spells maxing out at 234 fire damage per round, the Wall of Force has precedent for not allowing extreme temperatures to affect the enclosed creature.
As far as heating the interior by Radiation,
Nothing can physically pass through the wall.
That would include things on the electromagnetic scale, since the waves would physically need to pass through the wall, and light would be an exception (not the rule), since the wall is specified to be invisible.
The rules are unclear for the lich.
The Lich's "gaze effect" is this:
Frightening Gaze (Costs 2 Actions). The lich fixes its gaze on one creature it can see within 10 feet of it. The target must succeed on a DC 18 Wisdom saving throw against this magic or become frightened for 1 minute. The frightened target can repeat the saving throw at the end of each of its turns, ending the effect on itself on a success. If a target's saving throw is successful or the effect ends for it, the target is immune to the lich's gaze for the next 24 hours.
You correctly observe that this is not a spell.
The lich has to fix its gaze on the victim, which suggests a more intense version of just looking at them. You can look at someone through a glass window, but can you fix your gaze on them? It's hard to say.
For other "gaze" attacks, it's more clear.
There are other gazes that use different language. The Medusa, for example:
Petrifying Gaze. When a creature that can see the medusa's eyes starts its turn within 30 feet of the medusa, the medusa can force it to make a DC 14 Constitution saving throw if the medusa isn't incapacitated and can see the creature.
The Basilisk is almost identical:
Petrifying Gaze. If a creature starts its turn within 30 feet of the basilisk and the two of them can see each other, the basilisk can force the creature to make a DC 12 Constitution saving throw if the basilisk isn't incapacitated.
Or the Nothic:
Rotting Gaze. The nothic targets one creature it can see within 30 feet of it. The target must succeed on a DC 12 Constitution saving throw against this magic or take 10 (3d6) necrotic damage.
Note all of these explicitly require the monster to see the victim (and sometimes for the victim to see the monster) but not to interact with it in any other way. By definition, a transparent barrier can be seen through, so they work.
All of these are called "[Something] Gaze", so I'd take them as establishing a pattern that should apply to gaze attacks in general, but that's a little less solid.
Best Answer
Not many.
Since Resilient Sphere is essentially a spherical wall of force, and wall of force breaks line of effect, you cannot cast through it. Any spells targeting an enemy through the sphere fizzle.
Summon monster should not be able to summon anything outside of the sphere. The caster needs line of effect to the point where the monster will be summoned, and resilient sphere prevents this.1
Psionics, if they count as spells, also cannot pass through, unless they specifically ignore line of effect (I don't have the occult book so I don't know much about Pathfinder psionics).
Hexes are a gray area. There is very little information on how they interact with magic, other than them being supernatural abilities. Since they should still follow the general magic rules, I believe they must also have line of effect.
Gaze attacks are specifically called out as working through wall of force, but I don't know of a way to gain a gaze attack off the top of my head. There are no spells that qualify as gaze attacks, either, as far as I know.
1 This is debatable, but probably how I would rule it. The question is, is this spell similar at all to teleport or dimension door? It could be considered either. On one hand, you are designating a location, just like Dimension Door. On the other hand, the range is much shorter. The major difference is, Dimension Door targets you, and you are already inside the sphere.