No.
Ability modifiers are applied to damage, but half-level isn't. So, in your example, the Flames of Phlegethos would inflict 3d10+3 damage.
Half-level is applied to attack rolls because they are ability checks. As a more general rule, feel confident to add half you level to every d20 roll except saving throws and to every static number which usually stands against a d20 roll (like defenses).
You're right. It's only base dice, no modifiers.
This probably sounds terrible, however, there is a good reason:
It's more balanced than it seems.
It's hard to tell sometimes; believe me I know. But, until you get more hands-on experience with the game, you have to give the game you're playing a "grace period" where you trust that the designers made the correct decisions, even if it seems like they didn't at first. Every campaign you play or DM is going to be different. If you change players or DM, then the campaign will be completely different. Some of those games, you will have players that built weak casters, and experienced grognards that built martial types. Although it is a roleplaying game, the "player skill" of the game comes in with decision-making during game time, and character building during creation.
That said, if you GM a game and you consistently encounter the same balance issues across encounters, then yeah, you should feel confident that houseruling whatever the issue is will improve your experience.
Your Wizard's spell slots are pretty big.
Bear with me a moment and imagine that the Wizard class is actually Batman. The biggest strength of the Wizard is that they have a tool ready for every situation -- at least, if they're a good Wizard. That's the player-skill-decision-making part. The spells your Wizard picked aren't bad, they just make him very good at things that aren't combat. Your Wizard has a ton of tools for dealing with non-combat situations. Imagine if Batman didn't bring his Batarangs when he went out on a mission. He's still the best detective on the planet, but now he isn't as effective as he would be in a fight. Spells are really, really good at solving specific problems.
That said, given the nature of the encounters you described, you might try throwing your Wizard a bone if you haven't already. D&D is more than just combat, and if you aren't giving your non-combat specialist Wizard any non-combat to specialize in, he'll feel like a useless player. If your player is concerned about his build, then you can allow him to switch out a cantrip, or give him a couple of good AOE scrolls in the next loot pile to scribe.
In addition, cantrips scale at certain levels. It's not immediately, but you can be sure that the cantrip will be a good standby at later levels.
Best Answer
Spellcasting Classes
A level 1 wizard casting ray of frost deals 1d4 + their spellcasting modifier in damage.
As you noted, a cantrip is automatically heightened to 1/2 of the wizard's level, rounded up. A level 1 wizard can therefore heighten something to 0.5*1, rounded up. 0.5 * 1 = 1. Therefore, the cantrip can only be cast as a level 1 spell (it's minimum level). As a level 1 spell, ray of frost deals 1d4 damage plus their spellcasting modifier.
A different way to think about this (but which reaches the same conclusion) is that ray of frost is a level 1 cantrip. A cantrip is a spell. Given that the highest level spell slot your level 1 wizard has is also level 1, there is no real possibility of heightening the spell. The cantrips page which you linked to says it this way:
Spellcasting Archetypes
If a character received basic spellcasting through an archetype, then this may be slightly different. In this case it is possible to have a cantrip heightened to a level beyond the spell slots they have available.
For example, a 4th-level fighter with the basic wizard spellcasting feat has a single level 1 spell slot. However, cantrips are automatically heightened to half their level - making them level 2 ( level 4 / ).