Deltora Quest I (the first series in the Deltora Quest collection) seems to match your description pretty closely (I know you said it isn't Deltora Quest in the comments, but hear me out).
The setting was in medieval times. There was no traveling dimensions. I can't recall whether there was magic or not. It wasn't on Earth, but on some invented world.
Deltora Quest occurs in a fantasy/medieval setting. Swords and sorcery and all that. No planar travel.
The goal for the characters was to collect a certain number of stones or gems (I believe there were 5 of them)...
The plot is centered on the protagonist's attempt to restore the Belt of Deltora by finding the seven magic gems that were scattered across the kingdom by the evil Shadow Lord.
I believe it was a trilogy, with a focus on the son of some nobility.
There are eight books in the series (one for each stone and one for the finale). It is revealed in the last book that:
The main character, Lief, is the true heir to the throne of Deltora.
I have some thoughts about the following specific plot points you mentioned:
At one point, they mention that if one of the guards in a country falls asleep on duty, he will be kept awake by his garrisson/brothers in arms for seven days. At some point the minstrel falls asleep, and the old fighter teaches him how to stay awake.
No idea about this one.
They travel through a forest full of leeches and other insects to reach one of the stones.
The first book takes place in the Forests of Silence. I don't remember leeches and insects but it has been a long time since I read these books.
Also, I seem to recall there was a member of an underground evil organisation among the nobility, who tried to poison someone and were able to do that because they built up an immunity to the poison.
The previous King of Deltora was betrayed by his top advisors in the first book, specifically Prandine, the chief royal advisor. It was a rather large conspiracy that resulted in the government being toppled and the Shadow Lord taking control of the kingdom. Prandine is killed during the prologue of the first book. I don't remember anything about poison though.
...(I believe there were 5 of them), all in certain colours. Every stone has a certain power attached to it.
The Deltora series follows the protagonists' quest to restore the "belt of Deltora" with seven gems: Diamond, Emerald, Lapis Lazuli, Topaz, Opal, Ruby, Amethyst.
What is important about the gems is that they are unique. It's not part of a bigger magic system, but they are artifacts from somewhere.
The stones are powerful magical artefacts belonging to the Belt of Deltora, which was crafted to drive the Shadow Lord out of the Kingdom of Deltora. Each stone has unique magic powers. For example, the Topaz has mind-affecting powers that give the wearer courage. The Opal can give visions of possible futures (though it is not always accurate). The Belt of Deltora's power can only be unlocked when it has all seven stones and is worn by the true heir to the throne of Deltora.
They build up a fellowship with some stereotypical fantasy builds: an old school fighter, a minstrel, a female archer and probably more.
The party starts as Lief (the teenage son of a blacksmith) and Barda (a grizzled warrior). In the second book they meet a druid/ranger-like character named Jasmine who can speak with trees and has animal pets. In later books they meet more companions who tend to re-enter and leave the party several times: Doom, Steven and Nevets, Dain, etc.
I believe it started out with one or two characters, but the group grew and every character had a certain colour stone at the end.
At the end of the series, once all the stones have been collected, the protagonists must assemble a member of each of the seven tribes in order for the Belt of Deltora to reveal the heir. Each gem corresponds to one of the seven tribes. Luckily for the protagonists, the Shadow Lord was considerate enough to scatter each gem in a different tribe's lands, so they have already recruited someone from each tribe. ;D