It looks like her A time could do with extending which could help her sleep a bit better in the day (one longer nap and less resistance to naps) and as a result more soundly at night.
At 9 months an A time could be around 3.5hrs, some LOs like one A time longer or one shorter so they are not necessarily all exactly the same length across the day. Often there will be one longer nap and one shorter too because with the longer A times there isn't always enough time in the day to fit in 2 long naps.
You might already have an idea of which A time and nap would increase more easily for your LO or to fit around baby groups or family circumstance, this isn't a one size fits all solution.
As example your day might look like this:
WU 6.30
A 3.5hr
S 10.00 - 11.30 (longer nap 1.5hr or 2hr)
A 3.5hr
S 3.00 - 4.00 (shorter nap 1hr or maybe just a 40 min CN)
A 3hr
BT 7pm
As your A times look to be around 2.5hrs at the moment I'd suggest you increase in increments of 15 mins every other day (or every 3 days) to avoid a big leap which can cause OT. If you get wake ups mid-nap you may need to resettle her to help her sleep longer or use a W2S (wake to sleep) if short napping is an issue.
think she is getting several teeth, but it is taking longer than 3 days as the book stated was normal.
teething can really drag on. Teeth can move without cutting for days, weeks or months before they finally cut through. The final cutting for my DS would be 3 nights or less but those few nights were *significantly* worse than the rest of the general teething. The general teething was bad enough, I didn't know it could get worse than that.
If you suspect teething pain you can try medicating at BT and see if it helps. Pain or illness always needs responding to and is not really a suitable time for sleep training although gentle training such as weaning a hand on the back is certainly possible. You will need to return to Lo every time there is a NW whether it is sleep training related or teething related to maintain the bond of trust. it can be exhausting but it does eventually pass.
Hope this helps