Since this question was asked, Jeremy Crawford has changed his mind on how this works.
In the April Rules Answers column for Sage Advice, Crawford has this to say:
Does moonbeam [or Spirit Guardians] deal damage when you cast it? What about when its effect moves onto a creature? The answer to both questions is no.
He goes on to elaborate:
Reading the description of any of those spells, you might wonder whether a creature is considered to be entering the spell’s area of effect if the area is created on the creature’s space. And if the area of effect can be moved—as the beam of moonbeam can—does moving it into a creature’s space count as the creature entering the area? Our design intent for such spells is this: a creature enters the area of effect when the creature passes into it. Creating the area of effect on the creature or moving it onto the creature doesn’t count. If the creature is still in the area at the start of its turn, it is subjected to the area’s effect.
This is the list of spells the article applies to:
- blade barrier
- cloudkill
- cloud of daggers
- Evard’s black tentacles
- forbiddance
- moonbeam
- sleet storm
- spirit guardians
Contrast with his original tweet, and follow-up.
PH Page 190 has a section on different speeds that I think would apply:
If you have more than one 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 ar less, you can't use the new speed during the current move.
Your Questions
If a creature with normal speed 30 entered the area after moving 15 feet, what is its remaining available movement?
You have a speed of 30 and move 15 feet. You then enter the area and you now have a move of 15 feet. Your already-traveled movement is subtracted from your move speed, which is 0 or less, so you can't use the "new" speed.
If a creature with normal speed 30/halved speed 15 starts in the area and moves 15 feet to exit the area, what is its remaining movement?
This is the opposite, you have a speed of 15, and move 15 feet and are out of the area. Your speed is now 30, and you've only used 15 feet, so have 15 more feet available.
Difficult Terrain
The section you are referring to is under the Speed heading, which states that "[t]he following rules determine how far a character or monster can move in a minute, an hour, or a day." Therefore, those rules wouldn't apply to in-combat movement since that has its own rules.
The rules for in-combat difficult terrain are found PH page 190:
Every foot of movement in difficult terrain costs 1 extra foot. This rule is true even if multiple things in a space count as difficult terrain.
Nothing about halved movement. Hence, I conclude that halved movement, at least in combat (since the spell in question has a duration of 10min, I don't see it being that useful outside of combat), means \$ \frac{speed}{2}\$.
Best Answer
Yes, Spirit Guardians' area depends on the size of the caster
Spirit Guardians has a range of self and the area of effect is described:
A distance of 15 feet is the same range that you could hit if you have 15 feet reach or a ranged weapon with 15 foot range. Unless you are using the variant facing rules you are assumed to have that reach in all directions from all squares that you occupy.
Therefore Spirit Guardians' area of effect is the area of a 15-foot reach from the caster. Assuming the caster is Small or Medium, this is slightly larger than a 15-foot radius, as it adds the square the caster is standing in. For a smaller creature that does not occupy a full 5-foot square, this is the same area as a 15-foot radius. For a Large or larger creature, the area is much larger.