How much movement does it take to stand up from prone when my speed is modified?
Answer: Half of your normal, unmodified speed. Not half your new speed.
This answers differs from the accepted answer. I'm almost reluctant to offer dissent against the one and only answer which probably enjoys general support, since there is no other answer. But because my answer does appear to match RAW, I will offer it up for consideration and I welcome comments.
The rule (Basic Rules, chapter 9, page 73) for standing from prone says this activity “costs an amount of movement equal to half your speed.” It could easily have said “…half your speed after applying any modifiers” but it did not.
Consider instead the rule (chapter 9, page 75) for the Dash action. It allows extra movement which “equals your speed, after applying any modifiers.” So in that specific case we learn that the modified speed is the amount to take into consideration. When the Designers wanted the modified speed to be taken into account they were clear about it.
I believe it is generally accepted that we should assume the words mean exactly what they say. “There are no hidden rules” is an oft-quoted phrase on RPG Stack Exchange.
I couldn't tell you the Intended Rules, but Written Rules appear to be: the cost of standing is half your unmodified speed.
So, to answer your first paragraph: with speed of 30, but slowed to 15, it costs half of 30 to stand from prone. 30/2-15 = 0. This would use up your entire movement for the turn and you would be unable to move.
That might seem harsh, but for what it may be worth, this interpretation appears to be close to how the 3.5 rules handled standing up while slowed. Barring any other abilities, if you used your "move action" to stand up you would not be able to move again that turn. I'm not claiming we should look at 3.5 to interpret 5e, but this is food for thought. Is there any reason to suggest Design Intent changed between versions?
When searching for Design Intent I found this Tweet from Jeremy Crawford, lead rules designer for DnD.
Question: “If you are under the effect of Haste and Knocked Prone,
does it take half your Hasted movement to stand?”
Mr. Crawford: “Standing up requires an amount of movement equal to
half your speed, not equal to half your potential movement for the
I am not entirely sure how to parse that response, but it appears Mr. Crawford is saying "No, you use half your unmodified speed and not half your potential (hasted/slowed etc.) movement in order to stand." After all, he could have simply said "Yes" if your new speed was indeed the number to work with.
You also asked:
What if I am also hit with a Ray of Frost spell while slowed, reducing
my speed to 5 feet for 1 round? Am I able to stand up at all?
Answer: Actually, your speed is 10 feet. (30-10)/2.
[Jeremy Crawford tweeted: “While the haste spell is active, it doubles your current speed. If your speed changes, haste doubles whatever your new speed is.” Accordingly, I think it likely we should assume Slow halves your new speed if hit by Ray of Frost.
Since standing from prone costs 15 feet, you are unable to stand up.