I am not a specialist of language.

I just wrote my impression.

I am looking forward to hearing

the others' opinions.

1.

"北の方では寒いです."

I guess there would be some preceding sentence.

for example,

"日本の大部分は温帯ですが"

Most part of Japan is located in the temperate zone,

"で" is some emphasis in this time,

opposite to the previous sentence,

though it's also OK without "で".

2.

"北には寒いです" is wrong.

3.

a)

When we don't specify the place,

we tend to add "方"

The north part is cold. = 北の方は寒いです。

Hokkaido is cold = 北海道は寒いです。

b)

In other cases, if the speakers are not

familiar with the place or the place

is far, they tend to use "方".

I have never been to foreign countries,

but Russia is likely cold.

外国に行ったことはありませんが、

ロシア（の方）は寒そうですね。

c)

Maybe depending on our culture,

we tend to make statement comparatively obscure.

for example,

- 3つ ほど（くらい）

- xxx, yyy, zzz など（とか）

Help : 北の方では VS 北には

You can't say "北には". You can say "北は寒いです".

"方" include meaning of "around".

"では", especially "で”　is emphasis subject. So it suggests southern area is not cold.

"には" stand for the place that something exist.

Nuance is little different, but you can say just "北は".