The NSAIDs I listed are veterinary drugs labeled for use in dogs.
Human OTC NSAIDS are generally not considered safe in part because an effective and safe dosage has not been established. This is expensive, which is a primary reason that species approved drugs cost more than unapproved drugs.
Human NSAIDs often work in dogs and cats. A number of years ago a client of mine used her prescription Naproxyn on her dog. It reduced the dogs pain significantly. She called me after a couple days because her dog didn't eat breakfast. Regretfully, the dog died a few hours later from a full thickness bleeding gastric ulcer. Alleve is the OTC version of Naprosyn.
Human OTC NSAIDs include aspirin, ibuprofen (Advil, Motrin), naproxen (Aleve) and ketoprofen (ORudis KT). Alleve is not safe. Ibuprofen is sometimes used by owners. An effective dose that has been mentioned is 5 mg/kg/day. Toxicity (gastric ulcers without other clinical signs) have been reported at 8 mg/kg/day. This is not an adequate margin of safety and sensitive dogs are likely to have more severe toxicosis at lower dosages given that drug effects vary within populations. Ketoprofen is approved in dogs at 1 mg/kg PO (tablets) q24h for 5 days. 0.25 mg/kg has been recommended for chronic therapy. Side effects are bleeding and kidney/liver failure, which are similar to NSAIDs in general.
Meloxicam is available by RX in humans and dogs.
For completeness, the commonly referenced aspirin dose in dogs is 5-15 mg/lb 2-3 times a day. I've always seen side effects well below the maximum dose and often below the minimum effective dose. Acetaminophen (Tylenol) is not a NSAID, but is sold OTC for pain and fever in humans. In dogs acute toxicity is generally not seen at dosages under 100 mg/kg/day. I've had clients use it compounded with narcotics long term for dogs in pain from terminal cancer. They all died of the cancer, not the acetaminophen. Toxic effects include liver necrosis, which can result in death.
Doxycycline dosage is 5 mg/lb/day. Side effects include nausea, vomiting, diarrhea, lack of appetite and photosensitivity particularly at higher dosages. In dogs, signs of acute toxicity are usually not observed unless the dosage of exceeds.
That is a summary of the NSAIDs etc. info I have. The numbers are from websites and a book titled the Pill Book Guide to medications for your dog and cat (1988).