For the Adventurer Trophy:
There are four multiplayer modes (Rescue, Team Deathmatch, Cry for
Help, Free-For-All) and you have to complete a match in each one. Note
that you have to finish a match and not just a round but if you join a
game that is already in session, that will count. You can also choose
either Ranked or Casual, it won’t matter.
Note: You can check your progress for this trophy and all other
multiplayer related trophies by choosing Feats & Setbacks from MP main
menu. They will be under Trials section.
That being said, you can check to see if each of the trophies are glitched if they say they are completed.
For the On My Way up Trophy and the True Commitment Trophy:
From what I have read, there haven't been many outcries about any issue(s) with this trophy, [not saying there isn't a local issue with your system] but be sure to check your status of earning this trophy as mentioned before.
You need 1,771,100 experience to reach level 60. Depending on your
skill level or whether you will boost it or not, it will take at least
15 to 20 hours to reach level 60. Also you won't unlock the trophy
until the end of your current round/match.
Source & boosting thread if interested in the Platinum.
Also note that you need to Prestige at least 3 times for the Platinum to pop.