No https://omniwallet.org is running its own core. So you don't need any other wallet anywhere to view your addresses.
No need if you are just checking your address.
Follow up question what wallet are you running?
Again your wallet doesn't actually receive anything, it simply reports on that address using the database behind the wallet and everyone is supposed to have the same copy of the database. For your wallet to report on that address correctly then it needs to be upto date (ie synced) The coins are sent to the address even if you never sync again your wallet.
If you want your wallet to see (report) the transaction then it needs to be synced otherwise the wallet's database does not have a record of the transaction. Everyone else does have the record and any omni-protocol aware wallet will tell your about the transaction.