Hi and welcome to BW
Looks like you've got a good routine going there, perhaps ready for a tweak now she's getting older.
At 6.5 months you'd be looking at a longer A time
http://babywhispererforums.com/index.php?topic=64158.0Around 2hr 45 mins would be a good aim for now, then working up towards 3hrs as she reaches 7 months.
You need to increase the A time just gradually over a period of time. As you are currently at 2 or 2.5hrs I would suggest working up in 10min increments, and with each increase leave it at that for 2 or 3 days, then increase again. This gives your LO time to adjust. You will also need to be gentle during the increase, your LO's body clock will be telling her it is time for her usual nap so some low key activity is needed to encourage her to stay awake a little longer. Don't worry her body clock will adjust to the new times.
You don't want to make the increase in A time too quickly as this can lead to OT (over tired), if you get a 20min nap this is a sign of OT, shush/pat her back to sleep to extend the nap and stay at the same A time for another couple days so she can adjust, then increase slowly again.
The increase in A time should help the nap increase in length, ideally you want 1.5 to 2 hrs for a restorative nap. It should also help reduce that long time falling to sleep as I think your LO is UT (under tired).
With two good naps coming just lightly later in the day you can also look towards dropping the CN (cat nap). It's normal to start reducing then cutting the CN at 6 months although no hurry if it looks like LO isn't quite ready. Don't do it until you get two good naps though or you'll have a very long A to bed and get OT. You might want to cap the CN at 30 or 40 mins for now then reduce again later.
http://babywhispererforums.com/index.php?topic=64837.0http://babywhispererforums.com/index.php?topic=85134.0To answer your questions
1. I think the playfulness might go once the A time is increased. LOs will fall to sleep earlier than they need to if their body clock is telling them a nap is due and if they are given the right cues (ie a wind down). Sometimes they fight a nap if they are UT but sometimes they just go to sleep and take a short, non-restorative nap. If she isn't crying I'd leave her to it, say goodnight or whatever your usual key phrase is and leave the room. Return if she cries though and sooth in whatever way needed (shush/pat in the cot or PU)
2. Again I think you will have better success on the falling to sleep with an increased A time. Don't worry too much about the shush/pat in the cot becoming a prop, if she needs it all the way to sleep then do so (even into deep sleep if she can't manage that alone). This sort of prop is not usually difficult to reduce and wean (slower rhythm of patting, a still firm hand, lifted but in position to start again if needed etc). In general you do as little as is needed to keep her calm and happy and help her to sleep. Even a fully independent sleeper will have times of illness, teething, developmental leaps and so on which disturb their sleep and we all return to this sort of soothing to help them back on track, no problem.
There's no problem with shush/patting for a 6month old, use whatever method helps to sooth her (preferably in the cot), it's just that some babies of this age get annoyed with the shush pat. It can be adapted to a rub or firm hand (on back, head, tummy etc) and a key phrase (eg 'sleepy time now') as the shush in particular can be frustrating for them (although very comforting to a new born).
3. I wouldn't worry about it taking her a while to fall to sleep so long as she is calm. This again is likely down to being UT.
4. I think that unless your nanny was to 'get into' the whole BWing thing herself it is very difficult to pass on all the different variables etc. The best you can do is to work out what routine you want and pass that info on to nanny. For instance to ask nanny to put her down to sleep later than she has been and to shush/pat to extend the nap if LO wakes earlier than 1.5hrs. Give instruction on the A time you are up to and when it is to increase.
You can also ask nanny to record the times for you S time, WU (wake up) time, and any info on extension etc. It's useful to know at what point extension is given (ie 20 mins into the nap or 45 mins into the nap etc) as this would help you know if LO is UT or OT.
As for putting down awake. All you can really do is ask her to do so and describe what you do at home. Beyond that it's kind of out of your hands unless you're prepared to get a new nanny if your wishes are not followed.
LO's are pretty clever though, they know who is putting them to sleep and will adapt to each care givers method. For instance my DS's grandmother will get him to sleep in her arms, he never ever sleeps in my arms and won't even do it if I ask him to because what he wants is his cot. Grandmother finds it very difficult to put him down my way although she has done a few times by following my exact instruction on time and what actions to take. She didn't think it would work but was amazed that she was out of the nursery within 2 mins and he went to sleep happily alone.
I would first extend the A, explain to nanny. Then after she's settled into that ask nanny to reduce the amount she holds to pat and instead put LO in the cot and pat. After a few days or a week ask nanny to try reducing the pat in the way I described above (she doesn't have to do it all in one go).
Hope some of this helps. Do feel free to ask for clarification or more questions should you need to.
And keep us posted on progress xx