Should Ugly Coins Be Fixed? Questions On Cleaning Coins.

Never clean coins.

Now that I've said that....

The decision to work on a coin must be a careful one. Your first coin looks like standard silver toning, something which will never "destroy" a coin except to leave the first few molecules of the surface a silver sulfide black. Yes, it can be removed, but the coin in question is so dark that the solution will harm the coin far more than leaving it alone. So that one doesn't get touched.

The second coin is another situation. It appears to have been exposed to chemicals, and that's likely already damaged the coin. No idea if it's progressive (unlike the final two), but as long as it's not a particularly rare/valuable piece it might bear exploration. Probably ought to.

The last two are verdigris-infested, and that is a progressive reaction which will inevitably eat the coin. It'll take a while, but it will. The Wing & Co coin might be so far gone that once removed you'll see the damage done, but the single spot on the last one might go away and leave nothing but an insignificant patch of color which will be a definite improvement. Learn about Verdi-Care for these, if you want to save them.

Explore the cause of the color, the value of the coin, and the relative ferocity of the necessary conservation process. In each and every case. Some will qualify for conservation, the vast majority will not.

Note, the terms "cleaning" and "conservation" are interchangeable in a sense - it's only the need of the coin which determines which term to use. One will make the coin worse for future generations (you're not a "collector," you're a "temporary caretaker"), and the other will make the coin better for posterity.

I've never picked up a coin with a glove on. I refuse to surrender any dexterity whatsoever when handling a collectable coin. Of course, I know how to handle coins, too - I'll be the first to slap someone I see holding a Proof in the palm of their hand.

for verdigris use bad-thad's VERDI chem / VERDI gone.

works wonders and cant possibly do any more harm than the green rot that is happening on the last 2. not sure what it will do for the white splotching but it wouldn't hurt to try along with an acetone rinse. I wouldn't touch the top coin.

A bit of disagreement. It sort of depends on the coin too. For example if a coin is worth only face value, you have little to loose by experimenting on cleaning. However, contrary to that is, although a coin may not be worth anything to one person, it could be the one needed by someone else.
With those presented I see nothing wrong with an attempt to improve them with a dip in Acetone. Might not do anything but will not do any damage either. If Acetone does nothing, then the next step is to attempt to stop whatever is causing what you see on them. At least placing them in a 2x2 flip will sort of make them less vulnerable to whatever is causing them to continue.
I've done some extensive attempts on cleaning such coins and in the end, basically ruined them. So many methods are posted on the internet on how to clean a coin and possibly posted by someone in an insane asylum. Actually so many people just post stuff that they have no idea what is real and what isn't.
As already stated, just don't clean coins.
Isn't it odd about cleaning. Don't clean a coin. Do clean your cloths. Don't clean old furniture. Do clean your car.
Constantly wash your hands. Don't clean an old Mummy from Egypt.

This.

How does an expert properly push a coin into an album or fitted holder without a glove?

I rarely, almost never try to clean a silver coin (possibly a quick acetone dip in some circumstances) however I frequently use Verdicare on copper coins particularly around mint marks to remove verdigris while searching for varieties etc. Verdicare works great but I encourage you to take ssuperdave's advice and do some research and learn when and how to use it properly.

An expert makes the hole bigger. The non-snarky answer is, with a fingernail around the rim.