The Boots of Speed WILL Allow you to Swim a bit Farther
If you lack a swimming/climbing speed, you use your current walking speed to swim/climb but must spend extra movement. The swimming/climbing rules don't shut off increases to your walking speed.
Speed is defined as Walking Speed
When you see the speed on your character sheet or in a monster stat block that is walking speed
Every character and monster has a speed, which is the distance in feet
that the character or monster can walk in 1 round. (PHB)
Also
Speed: A monster’s speed tells you how far it can move on its turn...All
creatures have a walking speed, simply called the monster’s speed.
Creatures that have no form of ground-based locomotion have a walking
speed of 0 feet. (DMG)
So if your character has 30 ft speed, that is a 30 ft walking speed. If they have a 30 ft walking speed they have a 30 foot speed. They are defined in the game to be the same.
Your Walking Speed Doubles With Boots of Speed
When you put the Boots of Speed on, your walking speed doubles. Per the above, that just means that the stat called "speed" on your character sheet gets multiplied by two. In the example above the OP's speed/walking speed started at 45 then becomes 90. This is his new speed.
You can use ANY AND ALL Speeds you have to Swim
Jeremy Crawford says:
The rule on swimming doesn't specify that you're using your walking speed. You use whatever speed you can to paddle, kick, and otherwise swim.
This means you can use your fly speed to swim. And your climb speed. And your burrow speed. As long as it is a speed you have, you can use it to swim. Of course, any of these that are not a swim speed will be used at a penalty (each foot of movement costs 1 extra foot), but you can still use them. Obviously this includes walking speed which is the default for most PCs when they go to swim.
Addressing the Specific Questions
Does OP's example character have >=1 speeds? Yes.
What type of speed(s)? A walking speed.
What is their current walking speed? 90 feet.
Can you use walking speed for swimming? Yes.
How far can they swim? 90/2=45 feet.
Simple as that.
In the example above, OP has a walking speed of 45. When he puts on the boots his walking speed becomes 90. THAT IS HIS SPEED. That is how far he can move in a turn. And according to the rules and JC's clarification above you can use that speed to swim.
To those that doubt this, what other number would the character use? His speed is 90 and he has no other types of speed. There is no other number he could use.
So, you would be able to swim 45' (90 feet / 2).
Combined movement
If you walk for 30 feet, you should have 60 feet of walking left as you said. You would then be able to swim for 30 feet (60/2).
Multiple Speeds
If I am playing a race or using an ability that grants me a swimming speed of 30, do the boots affect that?
Not at all.
Jeremy Crawford also said:
Bonuses/penalties to speed apply to your speeds in general, unless the text specifies walking, flying, etc.
Since it specifies walking speed, it only increases your walking speed with no effect on any other speeds you might have.
Example
Just to clarify things let me make an example where your character has a 45ft speed (walking) and 15 ft swim speed.
Putting on boots of speed would mean that you now have a 90ft walking speed and a 15ft swim speed. You could then swim 15ft using just your swim speed. After that you will have 75 feet of movement left since:
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. Whenever you switch, subtract the distance you've already moved from the new speed. The result determines how much farther you can move. If the result is 0 or less, you can't use the new speed during the current move. (PHB)
So with the remaining 75 feet of movement, you would be able to go 75/2=37 feet further in the water. This means you would have moved a total of 15+37 = 52 feet in the water.
What is the Purpose of Specifying "Walking" Speed in the Item's Description?
If the item's description had been written "the boots double your speed", then all of a creature's speeds would be doubled (walking, flying, swimming, etc.). This was clearly not the intent.
So, by writing "double your walking speed" they are specifically indicating that those other types of speeds are not affected by this item. Which makes sense since they are boots and not flippers or jet engines. It does not restrict how and when that walking speed can be used though. If they had intended that, they would have added language to that effect.
No, having natural armor is not wearing armor.
The quotes you give provide ample support for this:
you are ill-suited to wearing armor. [...] You gain no benefit from wearing armor
your speed increases by 10 feet while you are not wearing armor
It seems pretty clear that the tortle's shell is not "wearing armor" and thus wouldn't impede the Monk speed increase.
I'd certainly be prepared for a DM to rule the other way on this, but I don't think it's a problem, provided you treat the Natural Armor trait as a flat armor calculation of its own so that you're choosing between either 17 or the Unarmored Defense calculation of 10 + Dex + Wis. It means a tortle monk will have a slightly higher than average AC for a monk, but with their Wisdom bonus not adding to that 17, it should be fine.
Best Answer
The boots are applied to your speed, after other (permanent) additions
The boots give you speed, not movement. Speed is effectively the amount of movement you get per turn. Therefore the boots give you no benefit unless your speed is reduced by something else.
Because the movement speed from monk and from a feat are permanent and not optional or conditional in any way, your walking speed is 45 when you put on the boots so they do not boost it.