I think you can nominate a box. First you need to cancel the extra box subscription. After that you nominate your main box. If you then want to watch any subscription channel on the second box, you fire up the second box. I'll tell you that your subscription channels are on a different box, but invite you to nominate the second box instead.
So - it's a bit of a faff. And means you can only ever access subscription channels (and recordings of those channels) on one box at a time. But it's fairly trivial to switch between the 2.
No you can only have BT Sports on Sky or BT Box. I know I've been there.
I too find it ludicrous that if you have two BT boxes you cant have Sky Sports but I did some digging and its not actually BT's fault but Sky's. BT can't put it on two boxes as this imitates Sky's own "multiroom" and the agreement between BT and Sky is its a per box subscription so to put Sky Sports with two BT boxes they would have to charge you twice, one for each box and as BT won't do this you cant have Sky Sports with two boxes.
Way I solved it quite easily was to get a Now TV box and I just pay for a cheap pass off the Ebay when there is something on Sky Sports I want to watch !
Saves £££££'s and also does BT out of pocket too for the daft rule. Win Win.