[RPG] Does finding and removing a trap require one or two find/remove traps rolls

The reading of B/X leaves it open. In the period when it was for sale I can't remember a group that allowed a thief to roll twice. This is problematic at low levels because first and second level thieves have a lower probability of finding a trap than non-thieves. A 1 in 6 chance is 16.67% while first and second level thieves have a 10% and 15% chance respectively. For a character who is a specialist to be worse than non-specialists is problematic.

In the modern/OSR era the push for all characters to be able to try most things have revived the 1 in 6 which seems to have gone out of favor in the 3.x era (logically as it is handled by skills). In some cases these writers have argued that everyone has the 1 in 6 change to do thief skills (not just find/remove traps) but if a thief fails their skill is a second chance (or a saving throw). This is not the only or even most common interpretation from what I've seen but some OSR blogs champion it. One example is It's Okay, Gary Sent Us which is a revision of an earlier piece the author did for Fight On.

Doing some further research in my Dragon Archive CD, there is a Sage Advice answer in issue 62 about the chances of non-thieves having thieves skills and saying they do not. In general, the column was AD&D focused and without an explicit mention of B/X I don't consider it to overturn the 1 in 6 in an official manner. This point is emphasized by a question about contradictions between B/X and AD&D in issue 76 that said they were different and shouldn't be mixed. These were found searching Sage Advice in issues 52 (the introduction of B/X) through issue 80 (the end of 1983, the year Mentzer's boxed version of Basic came out). Based on the lack of an answer in Sage Advice in the relevant period I doubt TSR ever addressed the issue.

Checking for traps in combat does cost action economy

PERCEIVE SOMETHING Make a Perception check to perceive something, such as a hidden door, a concealed object, a group of creatures talking, or a monster’s tracks.

Action: Minor action. No action is required when the DM is using a creature’s passive Perception. Carefully searching an area (the creature’s space and squares adjacent to it) requires 1 minute or more.

DC: The DM chooses a DC from the Listen table, the Spot table, or the Find Tracks table.

Success: The creature perceives something. If the creature is carefully searching an area, it finds something, assuming there’s something to find.

This is important to keep in mind as there are quite a few minor action attacks that strikers get access to as they level and looking for traps is also costing him possible damage.

Out of Combat I would rule that searching precludes any other activities

The PC cant be actively searching for traps while carrying on conversations, etc. Passive Perception handles the normal awareness level of a PC while they move and do other things.

Talk to the player OOC

Express the issues you and the rest of the table are having. I can easily guess the having to adjudicate his search roles all the time just brings the game to a very slow grind. Maybe even make a social agreement about telegraphing whether or not traps might be in a certain area as a way to preclude the player searching all the time.