So the question here is:
Do bonuses to 'your speed' apply to flying speed?
Sadly, the answer is that is that it's not clear. I'll break down the arguments for and against.
For
- The 'Using Different Speeds' section on page 190 of the PHB says that "If you have more than one speed, such as your walking speed and a flying speed, you can switch back and forth between your speeds during your move.
- The 'Flying Movement' section on the same page says "If a flying creature is knocked prone, has its speed reduced to 0 [...] the creature falls [..]" which produces nonsensical results if 'its speed' refers to anything other than its flying speed.
- Unlike almost every other speed booster in the game, the Boots of Speed and the Boots of Striding and Springing both talk specifically about your walking speed.
- Crawford says that it does in one case, and in general.
Against
- Every other speed booster in the game just talks about 'your speed'. Some of them, like the Longstrider spell, are clearly only meant to be for walking speed.
- Mearls thinks bonuses to your speed only apply to walking speed. That said, he doesn't sound very sure, and he's not an official source of rules interpretation, so take that one with a grain of salt.
- Under the Speed section on page 182 of the PHB, it defines speed as: "Every character and monster has a speed, which is the distance in feet that the character or monster can walk in 1 round."
- Under the 'Your Turn' section on page 189 of the PHB, it includes this statement: "Your speed - sometimes called your walking speed - is noted on your character sheet."
To me, the last two arguments are the strongest ones, since they're the only attempts to actually define what 'your speed' means. That said, it's still a bit hazy, so I guess the answer (as ever is) has to be:
Talk to your DM. It's up to them to decide whether this works or not. I would definitely point out to your DM that this boost of 10 feet is effectively all you'd be getting from the Mobile feat, because:
- The second benefit of Mobile is pointless to you, since you're going to be flying all the time anyway.
- The third benefit of Mobile is equally pointless to you, since you're a flying Sorcerer. There's very little reason you'd ever get into melee, and even less reason for you to make melee attacks. (Although this works with melee spell attacks, so you can, for example, Vampiric Touch and then fly away without provoking opportunity attacks.)
Feats are meant to be powerful, and if all you're going to be getting is an increase from 60ft -> 70ft, it's not really a big deal. In fact, if I were you I probably wouldn't take Mobile. Haste is trickier, because it's a very small investment that would boost your speed up to 180ft (with the bonus Dash action), making you faster than any creature in the Monster Manual. Of course, it doesn't make much sense to allow one and not the other.
You've misinterpreted the movement penalty for grappling and the movement restrictions for jumping. They limit different things.
From the PHB, page 195:
When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
So you're not halving your jump distance, you're halving your movement speed. And you have to use that for jumping (PHB page 182):
Either way, each foot you clear on the jump costs a foot of movement.
This means that, for a player whose normal movement speed is 30ft, while grappling he can move 15ft, and that's also the maximum jump height he can achieve. Not even the Jump spell or the Boots of Striding and Springing will let him jump further than that. The effects, however, do let him get around the Standing High Jump penalty.
To summarize, when looking into how high you can possibly jump while grappling, you make the calculations in a few steps:
- Determine your movement speed. This is also your maximum jump height (distance).
- Halve that height if you're grappling a creature that's not two or more sizes smaller than you.
- Determine your base jump distance. Use your total Strength to calculate this (including any bonuses to Strength from items, spells or potions).
- Multiply the base jump distance depending on spells an items you have that increase your jump distance (like the Jump spell).
- If you're making a Standing High Jump, halve the distance from step 4. Otherwise, subtract 10ft from the height in step 2.
- Take the lower number between the numbers that you get in steps 2 and 4 (after modifications in step 5). This number is how high you can jump (and how much movement you'll have to spend to jump the full height).
You can use this little tool I whipped up to test out various combinations.
If you find the maximum jump height (step 2) to be restricting you more than your jump distance achievable (step 4), you can use spells and items that increase your movement speed to propel you further. The Haste spell is good to consider. While it is a substantially higher level spell, it lets you use an extra action to Dash, quadrupling your speed in total. The added AC is a nice touch, too. You can even combine it with the Jump spell, as the latter does not require Concentration.
Using just the effects in your example, however, the highest you can jump is 15ft, using a Standing High Jump. If you somehow throw in a Dash, you can jump twice as much. Note that, if the PC's base Strength is already 18, the Belt does not help increase the jump height in this case.
Keep in mind, though, one litle bit about the High Jump rules:
In some circumstances, your DM might allow you to make a Strength (Athletics) check to jump higher than you normally can.
The rule enables you as a DM to let your player jump beyond his maximum movement speed (this bit of the rule is written directly after the bit about the movement speed restriction). When can that happen, how much of a bonus does the player get, what the DC is and what the possible consequences are (chance of Exhaustion, for example) - all of this is up to you.
A commentary on the Belt of Storm Giant Strength
I personally dislike the use of this item. First of all, it is a legendary item, suitable only for the most glorious of adventurers. Giving this item to a player character that's not very high in level (18 at least, I'd say) would seem overpowered, and frankly uninteresting. Even at that level it might not be as good as it sounds. Surely, it's a fantastic item, but it makes all the precious Ability points a grappler had spent on Strength a waste. On the other hand, having the belt in possession early and spending the points somewhere else (or getting feats), the grappler's main mechanic would now depend entirely on the belt. Losing the belt would thus make the character almost useless. While you might not plan on having it taken away from him in any encounter, the player might become paranoid about it nonetheless.
Alternative to using the belt, using a Haste spell (3rd lvl) and the Boots of Striding and Springing (uncommon item), the player can reach 21ft with a Strength of 18, and 24ft with a strength of 20 (assuming a base movement speed of 30ft, and making a Dash with the Haste extra action) by making a normal High jump (with a 10ft running start). Throw in a Jump spell (1st lvl) and the High jump goes up to 50ft. Using these three effects, the Belt would only give an improvement of 4ft with a Standing high jump.
If the Haste spell is not available, and using Dash is not an option, then a better combo would be to use Boots of Speed (rare item) and the Jump spell. Just those two bring the High jump up to 20ft, which might be enough. In this case, too, increasing Strength above 18 brings no benefits (though I'm sure it does so in the other aspects of grappling).
Seeing as all the Belts of Giant Strength do provide great bonuses to almost all aspects of grappling, I would award the legendary version at lvl 20 as a special Grappler capstone. Of course, not without effort. It's a nice excuse to lead your players into a perilous dungeon. I would also house-rule that the belt require a Strength of 20 to wear, just so all those Ability points don't feel like they're going to waste. If your player left his strength at 18, though, that might not be necessary.
At lower levels, a potion of Giant Strength (whichever tier you fancy) might be a suitable reward for a heroic deed, and since it's only temporary it wouldn't break anything (including fun).
Best Answer
Jumping is limited by your normal movement speed
The rules for jumping include this very important clause:
This means that you can't jump farther than you can move on a given turn, regardless of your maximum jump distance or height. So, if you have a speed of 30 feet, it doesn't matter what your maximum jump distance is, you can only jump at most 30 feet, either horizontally or vertically. Any effect that increases your speed or otherwise grants you extra movement on your turn (including the dash action) will allow you to jump farther, up to the limits you have calculated: 120 feet for a long jump or 60 feet for a high jump. However, to achieve these maximum jump distances, you will also need 10 more feet of movement to get a running start. So if your maximum jump distance is 120 feet, you actually need 130 feet of movement to make that jump.
Tempestuous magic won't help you
Unfortunately, it seems that the storm sorceror's tempestuous magic feature will probably not help you jump any farther. The feature says:
As explained above, what you need is increased movement on your turn, and not only does this feature not provide that, it prevents you from dashing, since you need your action (with rare exceptions) to cast a spell in order to activate the feature. Unless the spell you cast more than doubles your speed, you would be better off taking the dash action instead of casting the spell. So, it seems that this feature is more useful for evading opportunity attacks than for extending jumps.
Without tempestuous magic, by my calculations, your maximum long jump distance will be 126 feet, and your highest high jump will be 45 feet (measured from the ground to the bottom of your feet; add your height as necessary). Both of these assume a 10-foot running start.