Theoretically you could put a different LiPo battery in. The charge circuit is set up for 500mA max charge rate so obviously the charge time would be longer.
However, I would have thought you would struggle to find one with much higher capacity that would fit.
I seem to get a pretty usable lifetime (3 hours) out of the regular battery which equates to a good working interval for me as I switch it off when not in active use. You can help here by turning the backlight down to the lowest level you are comfortable with. I also have a back up USB 1300mA battery unit which I can plug in to DSO USB if I ever run out.
Yes. It is good to have some extra probes. The Quad has 4 inputs 2 analog, 2 digital plus the signal source so one can need up to 5 connections to make full use.
Those linked to are good in that they have the right connector to plug straight in to the DSO. Many standard probes have BNC connectors and would need adapters.
Few other points to consider here. Normally you would use 1:1 mode when using the digital channels and certainly for the signal source. The 10:1 probes are switchable to 1:1 but it sometimes more convenient to have little digital clips for those and the signal source to reduce bulk when getting into small spaces. Like this
You can also get the mini-coax connectors for the Quad and make up your own probes for special purposes.
Also with a 10:1 probe you still need to be very careful if you intend to measure high voltages. This particularly applies if you have the metal case Quad where there would be more chance in touching something inadvertently.