I would try the battery from the one that works and just can't get on the web in the one that power cycles to determine if you just have a bad battery.

As far as getting on the web, go into settings>connections>connections>manage modem settings and make sure that t-mobile data is selected and shows connection as epc.tmobile.com. If it doesn't, choose edit, type that in as the apn and choose next then finish leaving username, password and the one other field blank.