Your speed and the distance you can jump are related, but separate.
The basic rules define a creature's speed like this:
Every character and monster has a speed, which is the distance in feet that the character or monster can walk in 1 round.
Other actions and features key off your speed to determine different modes of movement, or the way your movement works in combat. Most importantly, the description of movement in combat states:
On your turn, you can move a distance up to your speed.
Base jumping distance
The rules on jumping are provided here:
Your Strength determines how far you can jump.
Long Jump. When you make a long jump, you cover a number of feet up to your Strength score if you move at least 10 feet on foot immediately before the jump. When you make a standing long jump, you can leap only half that distance. Either way, each foot you clear on the jump costs a foot of movement.
High Jump. When you make a high jump, you leap into the air a number of feet equal to 3 + your Strength modifier if you move at least 10 feet on foot immediately before the jump. When you make a standing high jump, you can jump only half that distance. Either way, each foot you clear on the jump costs a foot of movement. In some circumstances, your DM might allow you to make a Strength (Athletics) check to jump higher than you normally can.
As you can see, jumping is not separate from movement; it uses your movement.
The jump spell
The jump spell (and the ring of jumping that you can cast it from) doesn't affect your speed at all. It solely triples your jump distance. What that means is:
- Instead of a running long jump letting you horizontally jump [a number of feet equal to Strength score] (and half that if you make a standing jump), the jump spell lets you jump [3 times your Strength score] in feet with a running jump, or [1.5 times Strength score] for a standing jump.
- Instead of a running high jump letting you vertically jump [3 + Strength mod] feet into the air (and half that on a standing high jump), the jump spell lets you jump [3 * (3 + Strength mod)] feet up with a running jump, or [1.5 * (3 + Str mod)] feet with a standing jump.
Note that the spell does not affect your speed at all. Jeremy Crawford clarifies here and here on Twitter that your jump distance is still limited by how far you can move (though you can take the Dash action to increase your available movement for the turn).
This one's easy. It simply doubles your speed until the end of the turn. Nothing more, nothing less. Anything that keys off your speed, including the distance you can move in a turn, is now keying off your doubled speed until the end of the turn. (Note that the trait does specify that once you use it to double your speed, you can't use it again until you move 0 feet on one of your turns.)
Tabitha the Tabaxi Bard, with jump and Feline Agility active
You specified this example:
Tabitha the Tabaxi Bard, strength of 10, is 30 feet from, for instance, a hell hound
If she has a Strength score of 10, her running long jump distance would normally be 10 feet (i.e. with a running start of 10 feet before she jumps); with a standing long jump, she could only jump half that distance, for a total of 5 feet. No matter how far she jumps, it uses her movement to do so (in addition to any movement needed for the running start if applicable).
With the jump spell, her jump distance is tripled. She could now make a running long jump of 30 feet; however, she would need a running start of 10 feet before she jumps to get that far, using a total of 40 feet of movement. (Without a running start, she could only jump 15 feet.)
However, tabaxi have a regular walking speed of 30 feet, so normally she'd need to take the Dash action (which adds an amount equal to your speed to your movement) in order to make a maximum-length running long jump. Thankfully, Tabaxi also have the Feline Agility trait, which allows them to choose to double their speed for one turn (and they can only use the ability to do so again after they spend a turn not using their movement).
By activating the Feline Agility trait, Tabitha has a speed of 60 feet for a turn; using her Ring of Jumping to cast jump on herself as a bonus action, her jump distance is tripled. She would thus be able to get a running start of 10 feet, jump a maximum of 30 feet (i.e. 3 * Str score), and still have 20 feet of movement and an action left on her turn to use as she wishes.
(That said, jumping across regular terrain doesn't really accomplish much, except maybe looking cool. She'd only really need to make that sort of horizontal jump to get over a crevasse/pit or environmental hazard.)