No creature is immune to any kind of damage unless its stat block explicitly says it is.

You can think of psychic damage as interfering with a creature's "nervous system" or whatever magical thing it uses as an equivalent. Even mindless undead have some way of perceiving their environment and then manipulating the various parts of their body to get their body over to it and attack it; psychic damage is essentially high power feedback (like the really loud screech when you get a microphone next to its speaker) on these channels. If something would be better classified as a trap than as a creature, then it's probably immune to psychic damage.

Radiant damage isn't really light so much as it is divine power. It happens to look like light, but it's not; a laser or high-intensity infrared source would do fire damage. Radiant damage is like a flashlight that's been imbued with holy power to accomplish a specific task (namely, destroying some poor creature); the holy power only does what the ability says it does (damage), so powers that do radiant damage can't be used for other purposes unless blinking a bright flashlight briefly could accomplish those same purposes.

A brief discussion of the other damage types:

• acid: read up on how acids work for more, but basically it reacts with the target's physical substance and changes it into something else
• cold: removes (thermal) energy from the target
• fire: adds (thermal) energy to the target
• force: pushes on something without physically touching it (similar to, but not the same as, gravity & magnetism)
• lightning: adds electrons (lots of them) to the target
• necrotic: accelerates decay & the natural entropic breakdown of the target; some creatures might be immune to this if they can normally "live" for billions of years and they're made of something extremely stable, like gold (just because a golem's enchantment will last that long doesn't mean its physical form will)
• poison: causes undesirable chemical reactions inside the target
• thunder: compression wave in whatever medium the target is in; wouldn't work in a vacuum

The spell description does not indicate any resistances nor any immunities. It takes up some space and is made of force. All it needs to do is take one HP of damage and it drops to 0. How to target it would be situation dependent.

Therefore, all forms of damage cause damage to an Unseen Servant. The only form that is arguably not applicable is psychic damage, as the Unseen Servant is mindless, and no stat is offered for intelligence.

If it were within the blast radius of a fireball, game over.

Insofar as Cold damage is concerned, feeling cold isn't the issue. Cold damage being a damage type means that cold can damage it in terms of game mechanics. (Cold makes metal and glass more brittle ... and when we get to "does a mindless entity feel" we once again find that the rules are silent).

Bludgeoning, Piercing, Slashing. Acid, Fire, Cold, Poison, Necrotic, Radiant, Lightning, Psychic, Thunder and Force.

At least 12 of the 13 apply to it. Whether or not Psychic damage affects the Unseen Servant is a DM ruling.