Since I expect mining pool members to all agree what transactions to include and which timestamp to use, it can be possible that for the given block no solution exists.
That is usually not the case. With the stratum protocol which is most widely used, miners have no say as to what goes into a block except for the nonce and extranonce (extranonce is part of the coinbase transaction). So if a given block and all of its nonces result in no solution, the mining pool just constructs a new block with either new transactions, a different transaction ordering, or just a different timestamp and tells the miners to work on that.
As for the timestamp, it does not need to match real time at all. The timestamp just needs to be greater than the previous block's Median Time Past (median timestamp of the last 11 blocks), so the timestamp can be off of real time.
What if all 'big' miners choose blocks which do not have any solution? Could the network stall?
A miner is not locked into only working on the block that they chose. They can construct another block and work on that. The network does not know nor does it care if a miner had to construct a different block to mine.