Before I went to Valak Mountain, I completed every quest that I have that I didn't do, except "Paola and Narine", which requires a good affinity between two female characters. At this point I have Sharla and Melia, but they haven't reached the next stage yet. In fact, Melia only have a good affinity with Riki. The weather-based quests are also done due to a neat trick I learned somewhere: change the time and the weather might change. I got the "shooting stars" and "hot day" necessary to find the tough monsters to complete those quests.
One of the quests require reaching a tomb in Makna Forest, but it's guarded by a very tough monster that can insta-kill the party if it managed to use its powerful move. I used Melia, Shulk and Dunban because Melia can topple monsters fairly easily, while Shulk has a "Break" skill and Dunban has a "Topple" skill. The first time I managed to defeat that monster, while the second time I almost defeated it but it's gone for some reason. The reward is a book that promotes Shulk's Battle Soul to the Advanced level, while giving him a new Monado Arts skill - Eater.
In Valak Mountain, the first thing I did was to find Ice Cabbages, because it's a rare item that is required to upgrade Colony 6, and in the end I found 3 of these, which is good enough. The next thing to do is to advance the plot, something I haven't done yet.