These trophies can all be easily achieved at any time in the month, and most can be automated. Achieving a trophy is NOT dependent on any timing or placing in a high-score table.
Run the autoplayer in the Program Manager
Run the autoplayer in the Program Manager. The game cheats, so getting gold might take somewhere between a while and fuckin' ages. I only have bronze on my current account.
Autoplayer in the Program Manager. You need to win twice in a row to get gold, so it can take a while for gold on this one too. Silver is easy enough, though -- just five wins.
Autoplayer in the Program Manager. As easy as Cheat.
Autoplayer in the Program Manager. Bronze trophy is awarded after Tournament 1, but silver and gold aren't until Tournament 11, so they'll take quite a bit longer. Equally as easy, though.
I'm not sure if everything in this NQ2 speed guide is still current, but we still have an autotrainer available at the very least. Bronze is easy, silver and gold take quite a bit more patience but are certainly doable. This trophy certainly isn't quick, but at least it's safe.
Score-capped Flash games
Like the previous section, getting the trophy for these games is NOT dependent on placing in a HST. These Flash games have a maximum score, and every player that reaches that score will earn a gold trophy. These are relatively risk-free to score-send or AoB, since many players reach the maximum score. Even if a score is reviewed, if the score has been sent within a reasonable time there's no risk of being frozen.
Max score: 1687. AoB available.
Max score: 1250. AoB available.
Max score: 5000. AoB available.
Max score: 850. AoB available.
Max score: 1520. AoB available.
Max score: 1000. AoB available.
Max score: 40,000. AoB available.
Getting a trophy for these games is dependent on being better or faster than other players, but do not require a significant amount of skill or any risky score-sending.
Be one of the first ten people to reach the target score to get gold. The required scores for BTY are generally considerably lower than trophy or review scores, so tend to be safer to SS or AoB.
Quests can be completed every 12 hours, so you can do two a day. Have the quest page in one tab and the SW in another, and you can get 1980+ points per quest. Do it as quickly as possible twice a day until you get the trophy. Depending on how fast you are, it shouldn't take longer than a week as people quit doing quests as they get the trophy or just give up.
No one does petpet battles because they're fucking pointless and anti-fun, so you can usually get third place (or higher, depending on your petpet's level) by winning just one battle on the first of the month.
Easier if your account is older, but you can drop a few hundred thousand NP and hold F5 on the game screen and hope for the best. Again, best done on the first of the month.
Unlike Illusen and Jhudora quests, you're not guaranteed to get a certain amount of points. It's worth a try on the first few days of the month, since not too many people do these quests.
You can use the autoplayers in the Program Manager and ValorSuite to get up to the max level pretty easily, and you can usually get bronze at the beginning of the month at a much lower level.
At the first of the month, you can run the autoplayer in the Program Manager and have a decent shot at getting bronze.
Mostly for reference; maybe these games will be fixed in the future, but don't count on it.
Geos / Kacheekers / Armada
There isn't a lot of recent information about these games, but the trophies are definitely currently broken.