PvE
Demand for 4-man savage content has been conveyed to Yoshi-P at a greater rate recently. He wants to incorporate this feedback into the design of the new Deep Dungeon, perhaps by making the more difficult floors more readily accessible (as opposed to locked behind a time-consuming climb as in the current one). The new Deep Dungeon is planned to include new objectives and mechanics. Nova Crystallis, Dengeki
The Sightseeing Log will include 45 new sights at launch, but there is also a Hard Mode version planned for 4.x Ethys
Cross-world friendslist and linkshells again confirmed to be in the works MMORPG.com. More "world-less" systems planned - they are considering cross-world FCs Avalonstar
FC crafting update in 4.1 Nova Crystallis [My comment: The question was about "personal airships or aerial combat" that was mentioned as an idea before Heavensward. Yoshi-P basically says "No comment yet, but we have something coming soon." We know submarines are coming to FC crafting, but could there be more?]
New Aquapolis again confirmed to be coming in patch 4.0 Easy Allies
Stormblood story is split about 50/50 between Othard and Ala Mhigo. There might be a few more percentage points in Ala Mhigo actually. Avalonstar. There was a significant part of the Lochs map (the area that features Ala Mhigo) that people at the media tour were not allowed to access due to spoilers [MrHappy].
Some sort of spectator mode for PvE raids again confirmed to be coming in the near future (will be more involved than just spectating) Easy Allies
The final battle of Omega will be as visually impressive as T13 Bahamut. Yoshi-P hints at something especially impressive for the Savage version. MogTalk - Frosty and Layla
"Super Savage"/beyond-Savage content in odd-numbered patches reward bragging rights (more than a title - loot). High end versions of raids we've already seen. Will not be a full raid tier - will be 1 battle. Difficulty similar to Second Coil Savage. Same designer as the Second Coil Savage. MogTalk - Frosty and Layla [My comment: I've speculated before that this will be 1 old 24-man Savage boss. They've already shown Ozma Savage footage and they've talked about Ozma Savage on multiple occasions... so maybe?]
Edit: "The current plan... is odd numbered patches will have one dungeon and some other new content while even numbered patches will have two new dungeons and QOL improvements." MMORPG.com [My comment: This wasn't presented as a direct quote, so it could be a misunderstanding (especially since the following paragraph explains that Yoshi-P doesn't want to add older dungeons to Expert roulette for fear of people abandoning if they get a higher ilvl one), but it kinda makes sense with how they plan to focus on more diverse types of content, such as releasing the beyond-Savage boss, the siege PvP mode, the GC Squadrons content, the FC crafting update, and likely Eureka all in 4.1. Then a new Aquapolis in 4.0, a new Deep Dungeon eventually, and a new content category in 4.2 and beyond]
PvP
New PvP mode was initially aimed for 4.05, but it took too long to develop, so it's planned for release in 4.1 instead Nova Crystallis, PlayStation Access, PCGamesN
The new mode will involve large-scale battles between 2-teams. It doesn't really fall into either of the Frontlines or Feast categories, so it's something a bit new PlayStation Access
It will involve mechs and weapons - something about a certain raid boss (Brute Justice was mentioned) 4gamer, Famitsu, MogTalk - Frosty and Layla (timestamp 31:15) [My comment: This could be the siege/castle raid mode they talked about on a few occasions in the past here and old concept art here. With mechs? Heck ye!] Edit: Very_Merri and MrHappy also confirmed siege weapon PvP in 4.1
PvP will be available from level 30 and as part of the free trial (ranked matches are not available to free trial players) Famitsu, 4gamer
PvP will grant EXP for your class in PvE from level 30 all the way to 70 Dengeki, 4gamer
Old achievements such as those from the Wolves' Den will be moved to the Feast. It seems they plan on removing Wolves' Den completely. Easy Allies
Frontlines rules might be put on some sort of a rotation (perhaps weekly) Easy Allies, 4gamer