Standing still, there is no way to tell via mundane means

Darkness vs Blindness

Being inside an area of darkness puts you inside a heavily obscured area. This is true whether the darkness is nonmagical or magical (as in the one created by Darkness).

PHB 183

Darkness creates a heavily obscured area. Characters face darkness outdoors at night (even most moonlit nights), within the confines of an unlit dungeon or a subterranean vault, or in an area of magical darkness.

All creatures inside a heavily obscured area suffer from the Blinded condition.

PHB 183

A heavily obscured area—such as darkness, opaque fog, or dense foliage—blocks vision entirely. A creature in a heavily obscured area effectively suffers from the blinded condition (see appendix A).

Silence vs Deafened

An area covered by the Silence spell makes a much more direct comparison to Deafness:

Silence

For the duration, no sound can be created within or pass through a 20-foot-radius sphere centered on a point you choose within range. Any creature or object entirely inside the sphere is immune to thunder damage, and creatures are deafened while entirely inside it.

And so, there is no way to tell.

Except...

You can leave the area of effect

Both Darkness and Silence create static, non-mobile areas of effect. An easy, mundane way to know is to walk in a straight line until you have walked a certain distance. If you are still blind, it's the Blindness spell; otherwise you will be able to see the area of darkness behind you. If you are still deaf, it's the Deafness spell; otherwise you will start being able to hear again.

Alternatively, you can cast Detect Magic

Blindness/Deafness is a necromancy spell, whereas Silence is an illusion spell, and Darkness is an evocation spell. If you could cast Detect Magic, you would be able to tell the difference between these schools.

You can also try casting spells

Spells which require a Verbal component fail entirely inside an area of Silence, but you can still cast them while Deafened. For example, you can use a Healing Word on yourself. If you regenerate hitpoints, then you must be under the Deafness spell.

Can you use Blindness/Deafness/Darkness/Silence as a tool?

On your final question: is it a tool you have as a DM? Yes, of course it is. You could set up traps and puzzles that require sight or hearing to be gone. Or you can have deaf party members due to the Deafness spell trying to communicate with the other non-deaf party members. It's fun (for you, at least).

Will the players feel hassled? That depends on them. If you would really like to spring it on them, I say give it a shot. But if you genuinely believe they will strongly hate it, ask them to be sure. The reactions of your players will depend on that particular set of players and is not something the people at this Stack can determine for you.

Your job is to present them with obstacles and puzzles to challenge them. This is just one of those ways to do that. So yes, it is within your toolbox if you want to use them.

Yes, readying a spell behind full cover would prevent counterspell

Counterspell depends on sight and a clear path to the target

Counterspell has a casting time of:

1 reaction, which you take when you see a creature within 60 feet of you casting a spell

That means that an opposing spellcaster must be able to see the intended counterspell target. Additionally, counterspell must follow the general rules that spells must also have a direct path to the target. Full cover should prevent either or both of these requirements from being met (depending on the type of cover used).

Readying a spell behind full cover and releasing it will prevent counterspell

Casting a spell while under full cover follows all of the appropriate rules.

You can cast a spell behind full cover even though you don't have line of effect per Jeremy Crawford:

The intent is that your target must be within range when you take the readied action, not when you first ready it.

And your trigger "when I have line of sight to the enemy mage" is perceivable.

Then, when you release the spell, it cannot be Counterspelled. Again clarified by Jeremy Crawford:

Counterspell foils the casting of a spell, not the release of a spell that was cast previously using the Ready action.

So, there is no reason why this would not work.

The trick will be in getting an available and convenient source of full cover.

Opportunity cost¹ and downsides

Since you are spending your reaction, that means that this strategy will prevent you from casting counterspell as well as any other reactions until the start of your next turn.

Also, you must concentrate on the readied spell which means that you will have to drop concentration on anything else you might have been concentrating on. Also it opens up the possibility that your concentration is lost due to opportunity attacks, enemy readied actions, etc.

Is it overpowered? No (Fool me once...)

At first glance this strategy might appear to be overpowered. However, like most strategies, it has counters. You might get away with using this once, but a smart enemy (especially one that has access to counterspell) will not fall for it for more than that. They can move to negate your cover, move behind cover themselves, prevent you from moving behind cover, and/or use readied action to enact any of the above and more. In the end, this strategy should only be overpowered if creatures aren't responding intelligently to it.

There are other ways to prevent counterspell too

It is also worth noting that making yourself invisible (which is not incredibly difficult at higher levels) or blinding the opposing caster would also prevent any counterspelling without many of the disadvantages of the readied action technique.

Additionally, if the terrain and spell allows, you can cast outside the 60' range of counterspell.

^{1 - Thanks @Slagmouth!}