To join the network, a Bitcoin full node will work its way through the network history, independently verifying the state of the network through time. So yes, each node will have downloaded and validated the entire transaction history, but a node can choose to use 'pruning mode', where it will discard unneeded transaction history after validation is complete.
The ever-increasing size of the blockchain is a heavy consideration when considering network and protocol updates. Bitcoin requires a distributed network in order to maintain its function, so it is important that the software can be run on inexpensive hardware, so that more people can participate.
ps: if i have to pay for a service using bitcoins, do i become a node of the blockchain ?
There are other ways to interact with the network. Many wallets are lite-wallets, which validate the chain via headers instead of the entire block. This is a less robust security model, but it works well for many people.
You generally do not need to pay a service in order to interact with the network (as a full node, a lite-wallet, etc).