No. You are never proficient in an ability like Charisma. Sorcerers are proficient in Charisma saving throws, so you add your proficiency whenever you must make a Charisma saving throw.
This proficiency wouldn't apply to a Charisma (Performance) roll, since that's an ability check, not a saving throw. You'd only gain your proficiency bonus on a Performance roll if you had a feature that gave you proficiency in Performance (or proficiency with the tool/instrument being used, or a feature that otherwise applied to the circumstances).
Only if you are ruled as using an improvised version of a proficient weapon
The relevant piece of information, PHB p. 147:
Improvised Weapons
In many cases, an improvised weapon is similar to an actual weapon and
can be treated as such. For example, a table leg is akin to a club. At
the DM’s option, a character proficient with a weapon can use a
similar object as if it were that weapon and use his or her
proficiency bonus.
An object that bears no resemblance to a weapon deals 1d4 damage (the
DM assigns a damage type appropriate to the object). If a character
uses a ranged weapon to make a melee attack, or throws a melee weapon
that does not have the thrown property, it also deals 1d4 damage. An
improvised thrown weapon has a normal range of 20 feet and a long
range of 60 feet.
If you use an object that's similar to a weapon as if it were that weapon, that assumes you're using the weapon it was intended, with all of the applicable damage types, proficiencies, and properties associated with it.
Alternatively, if you use an object that bears no resemblance to a weapon, the object can deal 1d4 damage and the Thrown property.
However, you cannot use an object that's similar to a weapon and use it as if it were that weapon, and still have the object be something that bears no resemblance to a weapon.
If you attempt to throw a weapon without the thrown feature, you are no longer using the weapon the way it was intended, and it has to be treated as a different weapon. If a different weapon is applicable, use the new weapon's properties and proficiencies. Otherwise, treat it as an Improvised Weapon, with the applicable properties and proficiency.
With this information, you effectively have two choices:
Have your DM treat the original weapon as a second, different weapon that has the Thrown feature. Example: treating a Halfling's War Hammer (no Thrown) as a Goliath's Light Hammer (with Thrown).
Make your attack as using an Improvised Weapon, dealing 1d4 damage and using the default thrown option of 20/60 range, and only adding proficiency if provided by another feature (as from Kensei or Tavern Brawler)
Best Answer
Ignore the last statement about longsword proficiency
This is a bit of a guess, but I'm pretty sure that last additional statement about longsword proficiency is a side-effect of how D&D Beyond works. When you create a magic weapon or armor, you have to specify a weapon or armor type that your item is based on.
An additional sentence is then added automatically to the description stating basic proficiency rules or appearance.
This is actually a problem for items that specifically state special properties in the description that overrule general proficiency rules or appearance - the Sun Blade being one such example, where proficiency in either longswords or shortswords allows you to be proficient in using it. (And I've hit the problem myself when creating homebrew items!)
The same item in the Dungeon Master's Guide book (page 205-206) does not have this additional sentence, lending evidence that this last sentence is just a "D&D Beyond implementation issue".
@Someone_Evil provided this link to the Sun Blade in the DMG compendium for comparison. It does not include the second portion you quoted; that is copied directly from the entry for the regular longsword.