Hello, my name is***** and I have over 20 years of experience as a veterinarian. I am sorry to hear about Archie's intermittent lameness and gagging. I cannot connect the two problems with any one specific disease process.
The most common reasons for gagging in a young dog are infectious, such as a respiratory infection (Bordetella or Kennel cough) or a history of chewing and ingesting things he should not, or an overlong soft palate.
It is very common for dogs to limp more at a walk then when running as running allows them to briefly bear weight, and not always full weight, on a limb whereas walking puts full weight on an affected limb. In a young dog lameness is often related to overuse injuries (often soft tissue or growth plate issues) or secondary to congenital joint and growth abnormalities.
Which leg is Archie lame on? There will be different disease processes that affect the front legs versus the rear legs.
Have any radiographs been taken? If not since this has persisted I believe they should be taken now, or repeated to look for changes?
Any particular joints swollen?
Is he most lame after sleep or after heavy play?
If you can give me more information about the limp and which limb is affected I can be a little more definitive.