Generally speaking, the difference you have described is true. To be precise, it depends on the environmental parameters. What you said is true for constant pressure and constant temperature. For constant temperature and constant volume we would have $\Delta U$ instead of $\Delta H$, and Helmholtz free energy $\Delta A$ instead of $\Delta G$.
But the case you describe, the open beaker, is the standard case and your description is correct.
If a reaction is "spontaneous", it does not necessarily mean that the reaction actually runs at an observable speed, which is a totally different question. Spontaneity is only concerned with thermodynamics. Even a thermodynamically favorable reaction might run so slowly that we would say they don’t run at all. One example is the decomposition of diamond to graphite.
2) That’s a little bit difficult since non spontaneous reactions do not run, we can’t experience them.
Some compounds, when dissolved in water, cause it to cool down (e.g. urea). These are examples of spontaneous but energetically unfavorable reactions.
Many biological reactions are energetically favorable but not spontaneous and need to be pushed or pulled by other reactions.
Generally energetically favorable equilibria ($\Delta H < 0$) will shift towards the product side if cooled. So the cold denaturation of proteins is one example. It’s not spontaneous at room temperature but is usually energetically favorable at room temperature.
Another example would be saturated solution of gypsum with a small amount of undissolved gypsum. No more gypsum is dissolved so the reaction is at the given conditions not spontaneous any more but if we would cool down the solution more gypsum would dissolve, indicating that this reaction is exothermal that is energetically favorable.
So the reaction of solvation in a stable gypsum solution with remaining solid gypsum is an example for a non spontaneous energetically favorable reaction.
Note that the last example is exactly the reverse of my first (spontaneous but energetically not favorable) example.
Of course the reverse reaction of any reaction that is energetically not favorable but spontaneous is an example of a reaction that is energetically favorable but not spontaneous. (see comment by Ivan Neretin below)