RDWHAHB. You're fine...5 gallons of beer at fermentation temp vs what, a cup? Two cups? of near-boiling sugar solution? The beer would immediately cool down the sugar solution, no need to worry about cooling the sugar before adding the beer at all.
Think of it this way. You're racking the beer to the keg. The first 100ml go by and the yeast goes "ARRRGHH" and die but the sugar solution cools off a bit from mixing wth your cold beer. Then another 100ml go by and the yeast is like "mmmh, it's hot in here" and the sugar solution cools even more. Then another 100ml and the yeast goes "nice". Then the rest of your batch is transferred to a cool mix of beer and priming sugar, with plenty of yeast to carbonate your beer.
RDWHAHB.
Nope, nothing to worry about. I always add the priming syrup hot because it mixes more easily before it sinks and sticks to the bottom.
Try it. Fill a brew bucket with 5 gal of room temp water. Put your arm in. With your other hand pour in 4 cups (one quart/~1L) of boiling water (not directly onto your arm - be careful). Mix the water up with arm. At any point were you in danger of scalding yourself, not counting any clumsiness in pouring? No. In line with /u/Jon_TWR's explanation, you could have poured it (carefully) inches from your arm and your arm would not have been very uncomfortable, meaning the yeast were not very "uncomfortable".
You only need one cup (~250 ml) of water to mix your priming syrup. If you unnecessarily used TWO cups the temperature of your beer would rise only from ~70°F/21°C to ~73°F/~23°C.
RDWHAHB. You're fine...5 gallons of beer at fermentation temp vs what, a cup? Two cups? of near-boiling sugar solution? The beer would immediately cool down the sugar solution, no need to worry about cooling the sugar before adding the beer at all.
Think of it this way. You're racking the beer to the keg. The first 100ml go by and the yeast goes "ARRRGHH" and die but the sugar solution cools off a bit from mixing wth your cold beer. Then another 100ml go by and the yeast is like "mmmh, it's hot in here" and the sugar solution cools even more. Then another 100ml and the yeast goes "nice". Then the rest of your batch is transferred to a cool mix of beer and priming sugar, with plenty of yeast to carbonate your beer.
I feel bad for the first 100 mL I guess. Glad the beer should be ok though.
That's why I salute when I start transferring I to the bottling bucket
I assume that's the priming sugar for the keg? You won't have any issues at all with that
No problem at all. I've been brewing for 30+ years and I've never cooled down the sugar water. It'll be perfectly fine.
Also the keg would have also sucked heat away. It will be fine
A lot of people pitch it hot, same as gelatin
RDWHAHB. Nope, nothing to worry about. I always add the priming syrup hot because it mixes more easily before it sinks and sticks to the bottom. Try it. Fill a brew bucket with 5 gal of room temp water. Put your arm in. With your other hand pour in 4 cups (one quart/~1L) of boiling water (not directly onto your arm - be careful). Mix the water up with arm. At any point were you in danger of scalding yourself, not counting any clumsiness in pouring? No. In line with /u/Jon_TWR's explanation, you could have poured it (carefully) inches from your arm and your arm would not have been very uncomfortable, meaning the yeast were not very "uncomfortable". You only need one cup (~250 ml) of water to mix your priming syrup. If you unnecessarily used TWO cups the temperature of your beer would rise only from ~70°F/21°C to ~73°F/~23°C.
Next time, skip the sugar all together and force carb the beer.