You can close the client in this stage of the trade. When you need to keep your client open is when you create an offer and you’re waiting for a peer to take it. There’s a popup which warns you when closing the app if there are open offers that won’t be broadcasted to the network.
To prevent scams, there’s no way to contact the seller. You can only talk to the arbitrator when a dispute is created (automatically when the trade period ends).
I’m not sure to catch all the points about your exact situation.
With which Bisq version did you engage in the trade ? v1.1.4 ? older version ?
Which Bisq version have you “also downloaded” ? v1.1.5 ?
I’m maybe too cautious, but, if it were me, I would stick to the same version for the duration of the trade.
It doesn’t matter which version you started the trade, there has not been any incompatible upgrades since version 0.5 more than two years ago.
You can close your client at that stage since you’re only waiting for the other party to complete the trade. You will have to open your client to complete it on your side later though. As long as your client is closed you won’t be able to know what has happened to the trade, that’s the whole point of the system, it’s p2p and no one else can listen in.