How to check the balance of each address or how we can look for balance?
You will have to scan the blockchain and find the UTXOs that are locked with your addresses in the locking script. You can either do that by running a full node by downloading and verifying the blockchain yourself, or run a Simplified Payment Verification (SPV) node and query other nodes using bloom filters. You can also use Electrum and import the addresses to create a watch-only wallet.
Did the amount move to the base address as like monero eg. the funds once send to the integrated address it will move the base address?
There is no funds "moving" to base addresses. Only the private key that can unlock the UTXO will be able to spend the bitcoins. Generally, when you import your HD seed into any standard wallet, it will start generating child keys and scan the blockchain to see if there are any UTXOs that can be controlled by these child keys.
How to restore the wallet?
Just backup the mnemonic or the seed that was used to generate the child keys and addresses. Any standard wallet will be able to generate addresses if you have the seed.