Right hope this helps, I think I've figured it out what you can and can't do. You should be able to do all 3 points you mentioned.
With both o2 and EE if your phone is on then you can use the watch fully without it being connected via bluetooth, which includes sending/receiving whatsapp message, emails. I have tested it up to 200/300m. So yes you could go for a run and leave your phone. You do have to fiddle around with a few settings to get it properly set up though, like remote connection etc.
With EE if the phone is off then watch will only work for calls, no other functions will work such as whatsapp, emails, texts etc. This is the difference between o2 & EE. EE you have this function but with o2 if phone is off the watch is useless. EE watch plan is Β£7, o2 is Β£5. The EE plan is easy to set up can do it all on your watch whereas o2 you have to go in and most times they don't know what they are doing.
Personally for the extra Β£2 I say go with EE. But EE tariffs are generally more expensive than o2, I managed to ring up and get a 30% discount and got 125GB, 12 months for Β£15. Also with EE you get 10gb on the watch and o2 only gets 1gb. Shouldn't be a problem, maybe if you listen to music whilst running that might use more data but I've not had long enough to see what an average monthly usage is
If any of that is unclear let me know, more than happy to explain and this has been a headache to get to the bottom of. Both EE and o2 being useless with their advice, and Samsung not being helpful either.
