[RPG] Can a 6th level rogue select Investigation for their Expertise feature even if their Intelligence modifier is +0

RAW the Rogue does receive the benefits of Expertise to his passive.

A passive check is a special kind of ability check that doesn’t involve any die rolls. Such a check can represent the average result for a task done repeatedly, such as searching for secret doors over and over again, or can be used when the DM wants to secretly determine whether the characters succeed at something without rolling dice, such as noticing a hidden monster. Here’s how to determine a character’s total for a passive check: 10 + all modifiers that normally apply to the check. - Player's Handbook P.174

The player has invested in making this character good at this, this is not a bad thing.

Rogues in general are supposed to be good at this type of thing, hence the proficiency bonus class feature and the player specifically spent a feat on making his passive perception better. The player could've spent this on another feat to specialize in another area (or gain a combat ability). This is a good thing, while it may make it difficult for the Rogue to be surprised by a trap or an ambush you should in no way try to outmaneuver this.

Don't worry, there are downsides to the Rogue for this.

To reliably spot traps and ambushes the rogue will need to be at the head of the marching order so that he has clear sight lines. Rogue's are not particularly hardy and the party as a whole is inviting more risk for this reward. Likewise while he is extremely excellent at spotting physical dangers, the Rogue will not be able to detect magic wards and other dangerous enchantments and may equally blunder into them.

Ways to handle his detection of a trap or an ambush

You are correct in assuming that only that PC has seen the trap/enemy. Unless he has some mode of telepathic communication he will need to speak out and announce the threat to everyone. The best way to handle this sort of thing is to pass notes or send text messages to the player(s) able to see/detect the issue and leave it to them to react and tell someone else. Intelligent enemies will see/hear the rogue warning the party and the combat should start immediately (no surprise round though).

The PC does not become omniscient of the trap upon detecting it.

The PHB itself is very, very vague on what information is received when a PC detects a trap. However, the DMG does have a nice little section about traps, their detection, and disarming them:

If the adventurers detect a trap before triggering it, they might be able to disarm it, either permanently or long enough to move past it. You might call for an Intelligence (Investigation) check for a character to deduce what needs to be done, followed by a Dexterity check using thieves' tools to perform the necessary sabotage...

...In most cases, a trap's description is clear enough that you can adjudicate whether a character's actions locate or foil the trap. As with many situations, you shouldn't allow die rolling to override clever play and good planning...

...Foiling traps can be a little more complicated. Consider a trapped treasure chest. If the chest is opened without first pulling on the two handles set in its sides, a mechanism inside fires a hail of poison needles toward anyone in front of it. After inspecting the chest and making a few checks, the characters are still unsure if it's trapped. Rather than simply open the chest, they prop a shield in front of it and push the chest open at a distance with an iron rod. In this case, the trap still triggers, but the hail of needles fires harmlessly into the shield. - Dungeon Master's Guide p. 121

Essentially though whether trap disarming is simply a dex check, a series of checks, and/or involves serious RP is up to you as a DM. I would encourage the open-ended approach the book suggests as it adds complexity and makes trap checking and disarming a more engaging process. Sitdown with the party OOC and discuss what the table as a whole thinks should happen for trap checks and move forward based on that consensus.

Reference the tool proficiency optional rules from Xanathar's Guide to Everything

You can't apply your proficiency bonus more than once to a single roll, as you've established; it seems like you should probably make this roll using your expertise in performance, since performance is a relevant skill and with expertise that grants you the best modifier. However, there are options to represent your particular combination of proficiencies beyond simply adding more bonuses to the roll.

Xanathar's Guide to Everything, in the Dungeon Master's Tools chapter, includes a section titled Tools and Skills Together, which addresses exactly this situation where a character might have overlapping tool and skill proficiencies. It makes a couple of suggestions about how a DM could adjudicate the situation:

Advantage. If the use of a tool and the use of a skill both apply to a check, and a character is proficient with the tool and the skill, consider allowing the character to make the check with advantage.

[...]

Added Benefit. In addition, consider giving characters who have both a relevant skill and a relevant tool proficiency an added benefit on a successful check. This benefit might be in the form of more detailed information or could simulate the effect of a different sort of successful check.

The chapter then goes on to describe the possible benefits of various tool proficiencies, and gaining advantage on a performance check incorporating that instrument is given as a benefit of proficiency with a musical instrument:

Skills. Every tool potentially provides advantage on a check when used in conjunction with certain skills, provided a character is proficient with the tool and the skill.

[...]

Performance. Your ability to put on a good show is improved when you incorporate an instrument into your act.

An extra benefit to a successful check might perhaps be allowing the bard an automatic success (or at least granting advantage) on a subsequent social skill check to influence the audience after the performance.

The intent of these optional rules from Xanathar's is to make tool proficiencies more valuable, in order to encourage players to take and make use of them as character options, and reward characters who have invested in such specialisation.