The Etherealness spell states, in part:

This spell has no effect if you cast it while you are on the Ethereal Plane or a plane that doesn't border it, such as one of the Outer Planes.

So no, the Outer Planes don't border the Ethereal, and furthermore the Etherealness spell has no effect when used there.

Unless otherwise specified, time passes normally on the outer planes (and any other plane)

As a general rule, you should assume that normal rules of reality apply except where it is clarified that they do not. The Dungeon Master's Guide describes the environment of various planes in the "Creating a Multiverse" section, and specifically of the Astral Plane on page 47:

Creatures on the Astral Plane don't age or suffer from hunger or thirst. For this reason, humanoids that live on the Astral Plane (such as the githyanki) establish outposts on other planes, often the Material Plane, so their children can grow to maturity.

This explicitly refers to creatures on the Astral Plane, and should not be extrapolated to other planes that are not the astral plane. If all of them had this kind of property, they would all state so in their descriptions, or it would have been clarified as a general property of the collection of planes in prior description. No such statement exists - so the conclusion is that time passes normally on other planes. Indeed, we're explicitly given that normally Astral-dwelling creatures will send children to live on other planes - often, but not exclusively, the material plane - so we must infer that time passes normally on at least some of the other planes of existence.

Referring to the 3e Manual of the Planes...

This is consistent with older material. In 3e, for instance, the Manual of the Planes handbook explores the matter of adventuring on the other planes of existence in some detail, and qualifies the different rules on different planes of existence by a system of "planar traits". The "timeless" planar trait, in this system, is a physical trait which is described so:

Timeless: On these planes, time still passes, but the effects of time are diminished. Those effects should be specifically defined for timeless planes. Such conditions as hunger, thirst and aging might not be affected in a timeless dimension...

In this material the Astral Plane is timeless, but the inner and outer planes are explicitly described as generally having "Normal Time", meaning time passes as it does on the Material Plane, unless specified otherwise by the plane's individual description. None of the outer planes described by the manual actually do so, so the normal passage of time is the general rule. Even the chaotic realms of Limbo and the Abyss apparently follow normal time (though the Abyss gets a mention that there are rumours of a layer where time actually flows backwards). Besides the Astral plane, the Timeless trait usually only comes into play when dealing with unique/custom demiplanes.

There's no particular reason to believe that this particular rule of the planes should have changed significantly between editions. As written, they seem consistent with each other - the Astral Plane is explicitly timeless/ageless, but time passes normally in most other places.