I would like to know if there is any sound quality lost by using a 6mm (1/4th inch) TRS jack adapter to connect into a 3.5mm (1/8th inch) TRS slot. I would assume there would be some quality loss because of the difference in sizes.
Assuming you are dealing with human audible audio transmitted as as a line level, analogue electrical signal, then No.
To put it in perspective, if your signal is one that cannot be transmitted through a solid piece of metal then you are going to already experience degradation in the wires.
If your signal is high enough frequency to require transmission down twisted pair or co-axial then such an adapter may be an issue, but would also likely not be used.
Where you may have issues are at the points of contact, e.g. bad connections leading to crackling noises if you fiddle with the plugs. Using an adapter means more points of contact and more points of contact means more opportunities for bad connections.
None whatsoever. Though I've heard that they can't take as much power for some reason. The reason we have 3.5mm plugs is because they allow for much smaller portable media as they take virtually no space to mention. They are, however, much easier to break - both connector and plug, they are harder to solder together as there are much smaller margins and stray heat can melt a lot of part you don't want melted if you're out of luck, and they are anything but sturdy. If you plug in a 6mm into virtually anything, you can be pretty sure it will stay there even with some load and tear on the cable. Use the same cable (and we do use these cables for a reason), and there is a much bigger risk of accidentally unplugging the damn thing.
Then we have Bantam connectors, also called TT (Tiny Telephone). They are a middle-size between the two, and have the sturdiness of TRS (6mm), but can be used in a much more space-saving way.
The 6mm jack mainly exist for historical reasons. Long ago phone switchboards used this jack to connect callers. This was standardized for audio and was used in studios and home HiFi systems. The 3.5mm jack was developed because there was need to reduce the size of the jack for portable audio such as the Sony Walkman.
Choosing one jack or the other has more to do with physical requirements than performance. 6mm jacks are much more secure and won't get unplugged as easily. 3.5mm jacks are more easily incorporated into casual electronics and smaller enclosures that can't accommodate a 6mm jack.
Many high end headphones even have a 3.5mm jack with a 6mm adapter that can screw on over the 3.5mm jack. This means they effectively use 3.5mm even with the adapter. This is enough to convince me that the quality between the two types matters very little.
There may be some signal loss from making another transition from one cable to another, but it would be no more or less than going through a 6mm extension cable. What can make more of a difference is the material used on the connectors to make sure that they get good solid contact, but surface area of the connection doesn't really matter, particularly since generally speaking, it only connects on one small point of the connector (per segment) anyway.