One the first question... I couldn't find any additional information about the required daily check in on their website. So I'll check in for the first few days and then try delaying my check ins to see what happens. That won't be too hard since my phone's on silent most of the day anyway. I'm more of an iPad / laptop / desktop / email kind of guy than a text messaging guy.
The app displays a running total of your mined Pi and seems to increase every 2-3 seconds. I've earned 5.7141 Pi so far in less than a full day. My guess is it stops mining after 24 hours until you tap the check in button to resume mining. That's how I'd program the feature - I'm a software engineer - but I would only be speculating. I'll know more later today.
In any event, what better way to keep you engaged than sending you a text message each day requiring you to tap a simple button. You'll never forget about Pi.
On the third question... The app did not ask my permission to send any other information than my name and phone number. I checked the permissions settings in my iPhone and there are no additional permissions beyond the usual Siri and Badges & Sounds permissions, and Background Refresh which I believe they need to keep the app mining Pi even when the app is closed.
They do limit you to one account per person, which I think they define as phone number. So if you have a bunch of phones, presumably you could create multiple accounts and mine more Pi.
I'll post an update as I discover more, especially