I have read that every once in a while, the new block is determined by a winning miner. The miner takes about 10 min to form a block out of the transactions he has received so far. Does that mean the network tells each miner all at the same time that it is the start of the race and that they are supposed to compute the next block? During this race does Bitcoin stop accepting new transactions?
No, block discovery is a memory-less process. Every try of discovering a valid block is completely independent from any other. Miners continuously search for a new block. The only thing that changes over time is which transactions they're trying to include and what preceding block they are building on-top of.
When some miner finds a block that fulfills the required criteria, they just publish it to the network. The propagation of the new block also works as the signal for all miners to switch work to a successor of this block instead of the predecessor.
Transactions are relayed in the network independently and continuously. Miners may update their block template to include new juicy transactions every once in a while, but it's up to them how often and when they do this.
Transactions are always accepted and included whilst the mining process takes place. The mining process itself is memory-less, therefore there's no "progress towards the next block".
Each time a new transaction is included in a block, transaction inclusion priority are recalculated based on fee rate (as miners are always incentivised to mint the highest paying transaction to reap the reward), then a block template gets produced with the highest-paying transactions included in a block and it is then distributed to workers to mine on.
Transaction selection and mining all take place simultaneously.