[RPG] How does Jack of All Trades interact with Reliable Talent

Yes, the Bard adds 1/2 proficiency to initiative.

In 5e all checks are ability checks. This is why every check in published materials is listed as Ability (Name), such as Strength (Athletics).

Initiative is a Dexterity check. Under Initiative in the PHB:

every participant makes a Dexterity check to determine their place in the initiative order. (p. 189)

If you can add an ability modifier to a roll, it's an ability check (A saving throw or an attack is a separate kind of d20 roll). If you have a Skill related to that Ability, and you are proficient in that skill, you add your proficiency bonus.

Initiative is a Dexterity check with no other Skill attached to it. Things that modify Dexterity checks work. Other things modify your ability checks as well - such as adding all or half of your proficiency bonus. The Bard has one example of this.

The Champion Fighter's L7 feature has another:

Starting at 7th level, you can add half your proficiency bonus (round up) to any Strength, Dexterity, or Constitution check you make that doesn’t already use your proficiency bonus. (BD&D 26)

Since Initiative is an ability check that uses your Dexterity, the Champion Fighter also gets half their proficiency to added to their initiative roll.

Yup, you halve-and-add your proficiency bonus to even passive checks.

Jack of all Trades. ...add half your proficiency bonus, rounded down, to any ability check you make that doesn't already include your proficiency bonus. (PHB p.54, emphasis mine)

Combined with

Passive Checks. A passive check is a special kind of ability check....

Here's how to determine a character's total for a passive check:

10 + all modifiers that normally apply to the check.

(PHB p.175, emphasis mine)

So as long your Jack of All Trades modifier normally applies to your Perception (Wisdom) check, then it applies to the passive version.