Personally I would cut the night feeds before looking at altering day time milk (with a view to increasing solids). If you start cutting the day milk now trying to increase solids you might just perpetuate the night bottles. At this age it is still the milk feeds that gets LO through the night, not solids, I know many people think increasing solids will lead to a longer night, but it's a myth.
LOs become interested in solids in their own time, some are straight in there like they've been waiting all their life for solids, others just aren't that interested for a while. It doesn't matter either way at this age, if she was approaching 1yo and still not interested then you might need to encourage more but 3 months is a long time in baby world so there is no panic or hurry just yet.
It is advised to begin finger foods around 6 months of age so at 9 months there is no problem with offering finger food in addition to the puree you are giving her, you might find she has more interest in finger foods. But even with these there is so much 'new' about it all, texture, taste, smell, temperature, that there is no problem if she just plays with it and mushes it up between her fingers to explore. Offer batons/fingers of steamed veg, toast/pitta bread, even meat. Ensuring she is properly sat upright and the pieces are long enough for her to hold and control. Offer a sippy or even a bottle of water along side solids to ensure she has enough fluid.
(as a side note, to show the contrast, my LO screamed at me for solid food from about 4.5 or 5 months, it was very hard to hold him off, and I know a couple of babies just now who are 8 months old but seem like much younger babies and have almost no interest in solids and are not yet sitting unaided - very different but still normal for their own individual needs)
WRT the snacking, when my LO began solids he suddenly needed a few more milk feeds in the day. Although he had already had his fill of milk at his regular E time, solids came an hour later (with sippy and bottle of water) and he HAD to have another milk bottle following the solids. It did seem like he was snacking but it was just a settling in period. I've seen threads with similar LOs needing an extra milk feed after a solids meal so I wouldn't worry too much about that.
And yes I think this is totally normal.
See where you are once the night feeds are gone and after introducing finger foods. hth