I know the (place)へ(noun)を(verb)に行きます construction, but I also know that in some instances the particle に can indicate the place where we're going. Is it possible to use に instead of へ in this construction? For example, to say 北海道に雪を見に行きます? If not, why?
Is it any different when there is no noun, just (place)へ(verb)に行きます?
In saying "going to a place", nearly all Japanese speakers, myself included, would use both 「へ」 and 「に」 interchangeably. That is the cold fact whether or not the strictest grammarians would approve of it.
Both of these sentences are correct and natural-sounding:
「北海道{ほっかいどう}に雪{ゆき}を見{み}に行{い}きます。」
「北海道へ雪を見に行きます。」
Saying that the two sentences mean different things would be nitpicking.
Is it any different when there is no noun, just (place)へ(verb)に行きます?
No, it is the same. Either 「に」 or 「へ」 can still be used to indicate the destination.
It is perfectly OK to use two に's because the meanings/usages are different. One indicates destination and the other, purpose. For instance, you can say:
「北海道に/へ遊{あそ}びに行きます。」
「遊ぶ」 is the purpose of your visit. It just needs to be conjugated into its continuative form 「遊び」 to connect to the purpose-marking particle 「に」.
Selected answer is correct, but if you do want to nitpick the difference, it's that へ places slightly more emphasis on the motion toward the destination / departing from the origin, whereas に places slightly more emphasis on arrival at the destination itself. It's a bit like the difference between a visual where you draw the arrows indicating going to Hokkaido from here like this:
here -----> . . . . . . . . . Hokkaido
versus like this:
here . . . . . . . . -------> Hokkaido
For へ and に, respectively. In complete sentences, this basically never makes any difference except in how a sentence feels.
However, by the very fact that に has multiple possible meanings here whereas へ only has one, へ is preferred as being clearer when context is low, such as incomplete sentences. For example, you could ask someone where they're going today by saying
今日はどちらに[行きますか]？
but since this could also be short for
今日はどちらに[しますか]？ ("Which will you choose today?")
it is much more common to say
今日はどちらへ？
Also, a travel agency will frequently use 北海道へ！ as a standalone sales line, but not so often 北海道に！ since it is ambiguous and also has less of a feeling of movement.
Here's another business example where へ is preferred: https://i.sstatic.net/Ldpsn.jpg (basically, "Come to us for rental property info in Okinawa")
Both are usable in this context.
に places more emphasis on the location, and marks the place as the final location, while へ marks the direction (you go to Hokkaido to see the snow, but you might continue traveling to another location afterwards). In my mostly textbook exposure to Japanese, I've seen the (place)へ(noun)を(verb)に行きます construct and would preferably use this in this context to, especially since Hokkaido is a pretty big place and can be narrowed down quite a bit. It'd be different if you'd specify longitude and latitude, then に would be most likely the proper particle to use - but then again, how natural would that sound?
