This interview with Robin D. Laws might clarify some of the differences.
Basically, your investigators won't be denied essential clues throughout the course of the game. GUMSHOE prevents the awkward "Oh, you really missed that investigation roll. Your seasoned detective manages to completely miss any clues in the well-stocked library with a post-it note on one of the books that says 'READ ME'".
I really like the flavor of the Trail of Cthulhu book. Instead of giving you how many hit points Yog-Sothoth might have, it gives you several scenarios for what Yog-Sothoth might be, and what it's intentions may be. So even players intimately familiar with the Mythos may not know what is really going on.
Overall, it's a refreshing take on Call of Cthulhu, and is definitely worth checking out, even if you don't decide to use the mechanics.
The d3 is a rare die, and the d2 is a coin.
Substituting a D3 can be done with any die whose total number of faces can be divided by 3. These include in your case the d6 and the d12. The easiest way to do this is to use a d6 and say the following:
1 → 1, 2
2 → 3, 4
3 → 5, 6
Or in other words, divide by 2, rounded up. You can also work in cycles, with one number per facing before working in the next cycle:
1 → 1, 4
2 → 2, 5
3 → 3, 6
But this is less intuitive to use. The d12 can be used as a d3 as well:
1 → 1, 2, 3, 4
2 → 5, 6, 7, 8
3 → 9, 10, 11, 12
But this creates more possible results than a d6, and it's easier to use the smallest die you have. You could work in cycles here as well, but that makes things needlessly complex and I advice against using this:
1 → 1, 4, 7, 10
2 → 2, 5, 8, 11
3 → 3, 6, 9, 12
The d2 problem can be solved by either flipping a coin or rolling any die with an even number of faces. Either divide the results in half to determine the outcome:
1 → 1, 2, 3
2 → 4, 5, 6
Or make the even/odd split:
1 → 1, 3, 5
2 → 2, 4, 6
You can try using a d20 on the even/odd roll with all numbers above 10 being 2:
1 → 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
2 → 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
Or you can use the d6 or another die, whatever you like best.
There is no mechanical difference to doing it with a bigger die than using a d2 or a d3 unless the die you are using as a substitute is crooked.
You're not missing anything. Based on the description you provided, it sounds like your player purchased one of Q-Workshop's licensed "Call of Cthulhu" dice sets. Strange as it may seem, the license isn't to the roleplaying game necessarily, but to the story by H.P. Lovecraft and the associated Mythos iconography. The set of dice is decorated to look appropriate for Cthulhu-themed games, but isn't designed for a BRP Call of Cthulhu game — they're the seven dice usually found in D&D and other popular roleplaying games.