No, a monk can't use their Martial Arts feature with improvised weapons. Monk weapons are:

short swords and any simple melee weapons that don’t have the two-handed or heavy property.

Unfortunately, simple melee weapons are weapons that are in the "Simple Melee Weapons" category on the weapons table, and improvised weapons aren't present there. (Otherwise everyone who was proficient with simple melee weapons would be proficient with improvised weapons.)

So improvised weapons aren't simple melee weapons, and therefore they aren't monk weapons either and can't be used with Martial Arts.

Monk: Yes.

Monks can choose to use their attack's original damage or to use their Martial Arts damage dice. This often comes up since at lower levels, many monk weapons do higher damage than the Martial Arts dice can. As your quoted section says, monks can roll the Martial Arts damage dice instead of the normal damage.

Tavern Brawler: No.

Tavern brawler outright changes your damage dice for unarmed strike to 1d4. It doesn't have the language that Martial Arts does of allowing you to choose which damage amount to use.