While I am very sympathetic to the pangs of teething pain, I also find that after months of experiencing their children’s new teeth coming in, caring parents sometimes start to attribute symptoms to teething that may, in fact, be caused by something else. In your case, had you not included the first sentence about how your 10-month-old is having a lot of problems with teething, your description of his sleep disturbances fit those of a child who has difficulty falling (and staying) asleep on his own—a very common challenge that many parents are faced with and often mistakenly attribute to other causes such as (but not limited to) teething, colds, new foods, allergies, etc. The reason your description leads me to focus on his sleep rather than just his teeth is because of the pattern of your son’s waking and his hysterical crying, and because it sounds like your understandable attempts to soothe him by putting him in bed with you may well have added to his inability to fall asleep independently.
Normal sleep patterns for infants and children include several cycles through light sleep each night. Anything that makes it so a child awakens and can’t fall back asleep during these light sleep periods tends to result in the frequent wakings you describe. While teething itself can admittedly be irritating enough to cause a child to wake up and not be able to fall back asleep, this sort of pain is usually limited just to the day (or night) when a tooth is actually breaking through the gum. More often, infants who are or have become dependant on some sort of sleep aid (usually in the form of a breast, bottle, or being rocked/held to sleep) hit a predictable light sleep period during the night, wake up when they otherwise wouldn’t have, and can’t fall back to sleep because their sleep aid is gone. This sleep-depriving schedule then occurs repeatedly throughout the night.
So back to your original question—you certainly can offer your son some of the safe teething remedies such as a cold teether or a baby toothbrush to chew on. Also be sure to touch base with your pediatrician regarding the use (and dose) of a pain medicine such as acetaminophen if your son truly has a tooth in the process of erupting. But I also recommend that any parents faced with the sort of sleep challenge you’re describing make sure that they have in place a healthy bedtime routine, which may well resolve your son’s nighttime crying within a matter of a couple of nights.
And finally, you are correct about current safe sleep recommendations, as we know that babies not only sleep well when they learn to fall asleep on their own, but are safest when they are laid down to sleep in a safe crib without any pillows, stuffed animals, or fluffy blankets and on their backs on a firm crib mattress with a snug-fitting crib sheet in a smoke-free room.