Learn English – the difference between “stay home” and “stay home from work”

I'll address on this context and not other uses of 'beyond/across'.

In laypersons' language, 'beyond' has a connection with 'far'. So, if you are telling that someone is going 'beyond' something, you mark the hill as a point to describe what went farther.

OALD example says:

The road continues beyond the village up into the hills. -the village is the mark and the road goes farther into the hills.

On the other hand, 'across' talks about two points and in between is your object. So, if something is coming 'across' the hill, it's passing by the hill or getting 'through' it.

To get across the hill, you need great stamina. It's almost 45 km, you know!

Here, I'm taking two points -from where we are standing and I'm speaking and to cross the hill and go beyond it!

Somehow, I feel that 'came' fits better with 'across'; 'beyond' is generally observed with 'go'. However, there could be exceptions but what I said is 'general'.

"Check out" in this case means to go look, see, inspect, examine, etc.

e.g.

Check it out! There's a weird fish in the lake!

I need to go check out that sighting yesterday.

Check out the beauty of the falls!

"Check" simply is to inspect or examine or verify, among other senses.

http://dictionary.reference.com/browse/check?s=t

https://en.wiktionary.org/wiki/check_out

"verb + out" constructions are a type of phrasal verb, which are often idiomatic and their meanings are not connected to the first verb the same way each time. (You'll have to find out those meanings) English speakers use them a ton, gotta deal with that.

Often translating a phrasal verb will take up a different amount of words (maybe one word or so) in a destination language. (e.g. "to look for" is translated as "chercher" in French.)