No, a barbarian does not need to be in combat to rage.

Defining things as "Actions", "Bonus actions", "Reactions" etc. are only to specify how much you can do in a turn or round of combat, since you effectively only have a 6-second timeframe to do things. Much like casting spells that require a minute long casting time - you can still attempt to cast them... you just need to be uninterrupted for 10 rounds.

The only catch is that to sustain your rage, you need to attack (which doesn't have to be successful, mind you), or take damage, "every round", so a timeframe needs to be defined outside of combat to sustain it:

Your rage lasts for 1 minute. It ends early if you are knocked unconscious or if your turn ends and you haven't attacked a hostile creature since your last turn or taken damage since then. You can also end your rage on your turn as a bonus action.

^{Note: the rule does state "hostile creature", so it'll be up to you and the GM to define what states as "hostile". Spanking a misbehaving child, or punching a passer-by may not exactly qualify.}

However, as suggested in the comments, setting yourself on fire would be an ideal way to maintain the rage - constant damage, therefore, constant rage.

No, because the weapon isn't using Strength any more.

Down in the Making An Attack section (PHB p193-194) we get this...

Ability Modifier. The ability modifier used for a melee weapon attack is Strength, and the ability modifier used for a ranged weapon attack is Dexterity. Weapons that have the finesse or thrown property break this rule.

ALL melee weapons use Strength, by default.

The rule for the Finesse tag on p147 says...

Finesse. When making an attack with a finesse weapon, you can use your choice of your Strength or Dexterity modifier for the attack and damage rolls.

This gives you an exception....you may use Dex instead of Str when using a Finesse weapon.

the rule for a Monk says

You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons.

Emphasis Mine to show that using Dex instead of Str is optional for a Monk.

Notice how similar the phrasing is between the Finesse rule and the Martial Arts rule.

Specific over General

The fact that the general rule for melee weapon attacks is "Use Strength" with the specific exception that "Finesse Weapons break the rule," coupled with the fact that the Barbarian class description has to explicitly say:

When you make a melee weapon attack using Strength, you gain a bonus to the damage roll...

despite the fact that all melee weapons use Strength by default makes it pretty clear that, in order for the Rage damage bonus to work, you have to actually use Strength for your attack/damage roll. Otherwise, it would work with Finesse weapons used with Dex, too...and the call-out that it must be a melee weapon that uses Strength is meaningless.