I have been in your exact shoes with a 14 week old waking up even more frequently, sometimes up to every 15 minutes or twice in the same portion of light sleep. It was maddening, exhausting, and awful. In my daughter's case the frequent wake-ups were caused by gas pain (which did disappear around 15 weeks naturally), a poor sleep association with the pacifier, and a Moro reflex which lingered after the swaddle was dropped. As you know from having an older child the challenges of sleep at this age do eventually end, but it's a tough road.
14 weeks is right around the time, according to my pediatrician, when sleep associations begin to form. It is the beginning for some babies (others will start a couple of weeks later) of a major mental developmental leap during which short term memory begins to get a lot better, among many other things.
Mental Developmental Leap
Some children have a very difficult time sleeping during mental developmental leaps while others don't seem to notice them. Do you see other issues such as feeding difficulties, a general increase in fussiness, more crying, and so on? Those signs might indicate that your child has a tough time with mental developmental leaps. Mine had a very difficult time as an infant. Several of my friends' children did as well. But others' sail right through them without issue. So the developmental leap may be a factor outside of sleep assocations.
Biological Sleep Changes
Also right in the 3-4 month range there are biological changes to infant sleep. I'm not sure if you remember but I believe it is Weissbluth who talks about this the most clearly. You can find a brief overview at The Baby Sleep Site as well. Around now or maybe a little later you may notice that your child falls asleep faster - you no longer have that 20 minute period of twitching where you can't set your newborn down if you're holding him to sleep. However the partial arousals between sleep cycles become more distinct as the amount of active sleep decreases and the amount of quiet sleep increases (see the citations under the 4 month sleep section here). At this point his sleep associations come into play - he wants them and if he doesn't have them he'll fully wake-up and cry to get them.
Too Late Bedtime
Also according to Weissbluth and the experts-by-experience, though not training, at the Teaching Your Baby and Toddler To Sleep board, sometime between 7 weeks and 3 months is when baby's bedtimes naturally begin to need to move earlier. Bedtime, according to Weissbluth, must line up with a child's natural transition to nighttime circadian rhythms. Even with enough sleep a too-late bedtime may lead to overtiredness. I'd be concerned that the wake-up after the first sleep cycle has ended, 40 minutes into the night, is a sign that bedtime is too late.
The daytime nap issues will lead to nighttime issues. Daytime overtiredness can cause nighttime sleep issues. This age is very tricky for naps. Young infants this age are highly sensitive to time awake between naps being too short (so they are not tired enough to sleep through more than a single sleep cycle) or too long (so they are too tired and wake easily). Plus around this age many infants are dropping from 4 naps to 3 and are needing longer wakeful intervals but have trouble handling them. If your child is dropping a nap you can usually get away with 30 minutes to 60 minutes less total awake time during the day to help compensate for overtiredness. Make sure your child is not awake too long total - some 3 month olds can only handle 6 hours awake while others can handle up to 8.5 or even more. The total awake time includes all time awake, day or night. For more on fixing the 40 minute nap, see this answer I wrote.
Drop the Paci/Carrying/Rocking
Talk to your pediatrician about sleep training methods before doing them, but if you think holding or rocking to sleep and the pacifier are poor sleep associations you will have to address them at some point via whatever method you chose.
You might consider moving bedtime up by 20 minutes, holding that for 3 nights, and looking for improvement. Continue to do this if you see improvement until it doesn't help anymore or sleep begins to worsen.
Fix the Nap Schedule
Make sure that naps are happening with enough awake time in between for your child to be tired enough for a good nap (at least 2 sleep cycles or about 90 minutes), but not so long that your child is getting overtired. This chart would recommend a 1.5 hour interval of wakefulness between naps at this age. Consider keeping a log and seeing if nap changes help nighttime sleep. At this age I also am more of a fan of basing bedtime based on total awake time and allowing it to move rather than a set time. This lets you compensate for a bad nap day with an earlier bedtime and avoid overtiredness.
Stop Blaming Teething
While your first child got teeth early and you know the symptoms by now, you may not know since you didn't experience it the first time around that this age is prime for false teething symptoms. Children discover how to mouth things so they do it all the time just for fun. The mental developmental leap and overtiredness from poor sleeping cause fussiness. The ability to drool matures and extra drool happens even in children who are not teething. Plus studies are very conflicted on whether or not teething can cause sleep disruptions. If you are medicating your child against possible teething, with the consent of your pediatrician, I would blame other things first.
Wait it Out
Per the book Bedtiming, any efforts to fix sleep in the 4-5.5 month range will not work due to all of the developmental factors going on at this time. By 4 months if you don't have a good sleeper you won't until the developmental factors end. This is not everyone's experience whom I have talked to, but it was mine. We made many efforts to fix my daughter's sleep in the 3-5 month range. None of them stuck until she had passed her developmental changes and was mentally in a more comfortable place. If you find that this is the case with your child, there is light at the end of the tunnel.
Also since your child is not yet 4 months know that tummy troubles can persist until 16 weeks. Some childrens' digestive systems mature as early as 12 weeks per my pediatrician, but others take until 4 months. You should find that the impact of gas pains diminishes in the next few weeks.