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Long story short, I am currently running a campaign where there is a group in a dwarven mine that needs to do a time-sensitive mining operation. The group has about 6 dwarves with 14 strength each (and are equipped with miner's picks).

Are there and rules about how far one can mine in a day? If not, what is a good rule for this? How does strength affect this?

If you'll accept rules from older editions, the AD&D1e DMG has the numbers you need on p. 106.

To summarise, properly equipped dwarven miners can each remove 90 cubic feet per hour in very soft stone, 70 cubic feet per hour in soft stone, and 35 cubic feet per hour in hard stone. It also points out that with appropriate scaffolding, digging can happen simultaneously at floor level, and a few feet above that. It has no scaling for strength, but I might give a 10% bonus for all the miners being strong. Humans can only manage 75, 50 or 25 cubic feet per hour.

Given that the digging is time-critical, the tunnel will probably be as small as practical. I've been in a tunnel dug through rock under extreme pressure of time, the counter-mine at St Andrews Castle, which looks like this (picture from Wikipedia):

For scale, the trough in the floor is about a foot wide, and is where you're intended to walk; the larger width of the upper part of the tunnel allows for swinging a pick, or a weapon, and is 5-6 feet wide. The tunnel is only about 5 feet high from the bottom of the trough to the ceiling, and dwarves might well make it lower.

It was dug during the siege of 1546-47, when the besiegers started to dig quite a large tunnel under the walls. That tunnel is about 10 feet wide and 8 feet high, and was intended to allow planting a lot of gunpowder under the wall to collapse part of it.

When the defenders realised what was going on, they tunnelled desperately to reach the enemy tunnel before it got under the walls, which is why theirs is so small. They succeeded, just.

They didn't know exactly where to head, and were having to work by listening to the sound of the other side digging. That's why this tunnel has a bend in it; they made a mistake and had to change course.

For your case, if you're digging a small tunnel, I'd have two dwarves per shift, one digging as fast as he can and the other resting from his efforts. If you have other people around to help, they can clear away the rubble, saving the dwarves for the digging.

For a 5'x5' tunnel, 35 cubic feet per hour (dwarves through hard rock) is 1.4 feet of forward progress per hour, or 33.6 feet in 24 hours.

There's nothing specific in 5e.

As a DM, I generally refer to the Stone Wall spell for stone toughness, which states that a 6 inch 10x10 wall of non-magical stone has 15AC for sheer hardness and 30hp. As such, when the players want to tunnel through, I make them take attacks on the wall, with the wall chipping away 1 inch per 5hp, 10 feet tall and 10 feet wide. I treat mining picks as war picks for damage rolls, which base off strength.

For extended mining like you're talking about, nobody wants to roll for each attack over the course of hours. I'd get an average damage score from 2 characters (because 10 feet is enough space for 2 normal sized creatures) probably three rolls each. Any misses would count as zero, as the swing glances off the rock without doing damage. Average the damage and combine the two character's scores. 1 inch per 5 hp. I'd say that's the average distance they cover per round.

You'd also want some way to move all that rubble out of the way, so keep in mind carrying limits and the average weight of stone (150-200lbs per cubic foot based on stone type) so your miners don't mine faster than the rest of the party can clear the path.

I also apply the forced march rules for extended strenuous activities, but like everything else in this answer, it's all homebrew.