These are likely showing you that you have the resources required to upgrade your cards — congrats!
The blue arrow means you can ascend your troop — this fuses copies of a card into a slightly stronger version (so, for example, when you have 6 Common goblins, you can trade in 5 copies to be left with one goblin of Rare quality. A Rare goblin’s stats are slightly higher than a Common goblin’s, so ascending troops is generally a good idea.
Green arrows are on troops where you have enough traitstones to give them their next trait.
Neither has anything to do with the Underworld.
This is a Campaign tasks. Campaigns span 10 weeks, and the idea is that every week you have 2 gold goals, 3 Silver goals, and (I think?) 10 bronze ones. To get the rewards from the Campaign, you have to complete all the goals within the 10 weeks. Tasks from Week 1 will carry over to Week 2 if you don’t complete them “on schedule,” but nothing carries over after Week 10 (next week) so if you want the rewards you must complete all tasks before it is over.
The tasks are self-explanatory: if it says kill red troops, do so and the task will unlock, revealing the next one you’ve yet to complete if any remain.
Killing gnomes for pest control can be one of the trickier ones, because you can’t really predict when or where a gnome will spawn (except that they won’t be found in the Underworld, in training battles against yourself, in event-sigil battles, or in the Dungeons). Complete enough explore battles, though, and one is bound to show up.
Where you “are” in the game when you collect tribute (and thus receive your High Five Honor, if you got gifted one that hour) has no bearing on where it came from. The only way you can get Honors is from other players, meaning they have to have clicked on your profile somehow — people can only do that by seeing you in global, facing you in PVP, or clicking on your profile in the guild roster (which, since you don’t go to the first two much, is probably where your High Five came from)