The Boots of Speed WILL Allow you to Swim a bit Farther

Official Ruling by Jeremy Crawford

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.

3 miles per hour

Actions in Combat are not relevant to the travel pace rules. Rogues are faster movers in combat thanks to Cunning Action, but so are most characters as a character's walking speed reflects:

Every character and monster has a speed, which is the distance in feet that the character or monster can walk in 1 round. This number assumes short bursts of energetic movement in the midst of a life- threatening situation. (PHB 181)

[courtesy of MisterB in the comments]

Cunning Action and the Dash action do not affect travel pace. Compare this to a feature like the Totem Warrior Barbarian's Elk Aspect of the Beast which directly mentions travel pace:

Whether mounted or on foot, your travel pace is doubled, as is the travel pace of up to ten companions while they’re within 60 feet of you and you’re not incapacitated (see “Adventuring,” for rules on travel pace). The elk spirit helps you roam far and fast.

What about Longtrider?

Longstrider would increase your travel pace most likely. Travel pace of 3 miles per hour is derived from the common walking speed of 30 feet. According to the Dungeon Master's Guide:

In 1 hour, you can move a number of miles equal to your speed divided by 10.

Since longstrider increases your speed, your travel pace also improves during the duration (which by default is 1 hour).

Why can't I just Dash?

Combats rarely last more than a couple minutes and require a character to do a lot of things that would be very physically taxing should they be done for extended periods of time. In the rules for Chases from the Dungeon Master's Guide this is made apparent by this section on dashing (mentioned in HellSaint's answer):

During the chase, a participant can freely use the Dash action a number of times equal to 3 + its Constitution modifier. Each additional Dash action it takes during the chase requires the creature to succeed on a DC 10 Constitution check at the end of its turn or gain one level of exhaustion.

While this isn't directly related to travel pace it demonstrates that staying in combat mode for long periods of time (in order to Dash) could quickly result in exhaustion before you simply can't move further.