I'd like to understand the nuance differences between the following three ways
to ask a question. Let's say I am with a friend and thinking of going of the
beach. I can ask the following 4 ways.

>   1. 海へ行く？ [Standard way to make a suggestion]
>   2. 海へ行かない？ [Similar to the above but feels more aggressive]
>   3. 海へ行こう？ [Not sure if this is natural with a question mark. Removing the
> question mark turns this mostly into a statement which is somewhat forceful,
> though it would depend on tone of voice]
>   4. 海へ行きたい？ [This feels a bit too direct and not sure if it would be
> actually used in Japanese, or maybe only among close friends]
>

Above each I've written my comments about each of these, but they are just my
intuitive guesses.

If someone can tell me if I am on the right track and provide any more details
on the differences between these, I'd appreciate it.
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questions
Comparison of 4 different ways to ask something
These are just my personal observations, no reference material consulted, but
here they are:

>   1. 海へ行く？
>

I don't know how to comment on this one except that it is indeed one way to
suggest a trip to the beach. Hmm... but, used in isolation, it may sound a bit
blunt or out of the blue-ish compared to (2) because it doesn't have a
dedicated form for indicating suggestion/invitation.

>   2. 海へ行かない？
>

This would be what I'd say is the standard way to make a suggestion or
invitation, if there's any. ( _Why don't we ~ ?_ ). I don't think it sounds
any more aggressive than the first one.

>   3. 海へ行こう？
>

I find this one the most forward of the four. The speaker wants to go to the
beach and is in effect asking the listener to come along, or even take them
there.

>   4. 海へ行きたい？
>

This feels to me more like a plain old question than a suggestion. (If it is a
suggestion then it's an indirect one.) You are asking someone whether they
want to go to the beach or not.

* * *

Side note: In casual speech I'd drop へ in all of these sentences. Though when
writing I may not (if I were being careful enough), since 海行 in 海行く？could for
a split second look as if it were a word.
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I take four different expressions you gave in the following ways:

  1. 海へ行く？ Do we go to the beach / sea?
  2. 海へ行かない？ Shall we go to the beach / sea? / Why not go to the beach / sea? 
  3. 海へ行こう？ Let’s go to the beach / sea.
  4. 海へ行きたい？ Do you want to go to the beach / sea?

No. 2 sounds odds. We don’t say ”海へ行こう” in an interrogative form like this.
You don't need a question mark in this sentence.
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