This is NORMAL. She is a BABY. Babies change ALL the time, developing and what not and sleep patterns are NOT static. It will change lots from here on out, throughout childhood. Even teens have sleep trouble.
2) Per our Pediatrician, for the 1st year of life, a baby needs to be fed on demand. Yes, that's 24/7. Breastmilk/formula for the first year of their life, is the PRIMARY source of nutrition for them, NOT solids, NOT juice etc.
3) Their intake "needs" for breastmilk/formula will naturally ebb and flow, ACCORDING to THEIR growth spurts and development. So, this is why nursing/feeding them ON DEMAND is important... you must take THEIR cues. NOT GOING according to a "schedule." Otherwise, they will not be getting what THEY need, for growth.
4) Waking at 2:30 for a feeding is COMMON. Your girl to me, IS sleeping GREAT! Believe me. My kids at that age woke a LOT more... due to growth spurts, teething, and their increasing changes in their body. THIS is normal.
5) Do NOT try and "change" her and deny a feeding if this is what she needs. A baby KNOWS if they are hungry and will wake at night... she is ONLY 7.5 months old. It's normal.
6) as far as her napping... some babies just don't nap well anywhere else than at home. Some are just sensitive to what is around them... and if there are other kids around this may be keeping her "up" and then over-tired... and over tired or over stimulated kids do NOT nap or sleep well. They NEED TIME to "unwind"... just like an adult, right? Have YOU ever fallen asleep in 1 second as soon as your head hits the pillow? It often takes time... and even for adults to simmer down and get "peaceful" and in a "mood" to sleep. So... with a baby.. they NEED HELP to do this. It's normal.
7) If she IS napping when she comes home... and she is sleeping well at night (which to me she is)... then that's fine. But, perhaps the baby-sitter can try another technique to get her to nap there. ie: is it noisy there? Where does she nap there? is it pleasant? etc.
Believe it or not... your girl IS SLEEPING 'THROUGH THE NIGHT' if she is ONLY waking at 2:30a.m., assuming she is going to bed by around 8:00p.m. BECAUSE, for a baby 'sleeping through the night' is defined as 'sleeping for at least 6 hours straight.' Your girl IS doing this.
So I don't think there is a problem.
8) as far as solids... a baby will eat only what they can eat. At this age, they don't have to eat 3 meals a day like adults. At this age, they are JUST getting acclimated to 'eating' solids...it is only an 'introduction' at this point. Solids is a GRADUAL thing. Over time... over weeks...over months. So don't rush it.
If she is 'struggling' to eat her solids... then you need to take HER cues... she is telling you something. Hear it. Observe. Go according to the baby. Many times, a baby is 'rushed' to much and 'expected' to be eating like a toddler already at this YOUNG age. But no, they are NOT YET at this 'stage' of eating. So just hang-loose. Step back. Go according to your daughter.
AND, breast or bottle formula should STILL be her MAIN source of nutrition at this point anyway... on demand.
Don't worry. Your girl seems completely NORMAL to me, and sleeping well.
Keep in mind, that each month, and at each age-stage... they change and their needs change too... .throughout ALL childhood. Therefore, you must really cue into your baby... each baby is different. Their feeding like and dislikes change, their sleeping patterns, their habits, their activity levels, their physical and cognitive development changes... and ALL the "phases" they go through. Normal.
If you are "struggling to get her on a feeding schedule" as you say... then don't. Feed her on demand. And your babysitter.. .if she is also caring for other kids or older kids... she can't feed your BABY at the same times or the same methods. I was a baby-sitter once, and I fed the young babies on demand... and per THEIR needs, which is appropriate.. and I would try and 'duplicate' what the parents did at home, so there would be CONSISTENCY for the baby at my place and at home. And, their nap times, was PER the baby. Not "my" schedule. I would consider the Parents and what they said their baby's nap times "are."
If as you say, your baby does not seem to like baby food or eating solids....then she is probably not ready. Don't rush it. Wait. Just go slowly. And, a 'serving size' at this age is ONLY in terms of tablespoons or teaspoons anyway... NOT 'jars.' It is a very gradual process.
All the best, sorry for rambling,
Susan