Bitcoin transactions are broadcast publicly, and stored in copies of public ledger ("blockchain") maintained by clients such as yours all over the world. Even if your client is not running, or not up-to-date, you (or anyone else) can check if coins have been sent to a particular address, and how many.  Try blockexplorer.com or blockchain.info.

The only immediate reason you might need a client which is up-to-date is if you want to spend coins that belong to you. One of less immediate reasons to have the client running is that it helps the network by verifying and relaying new transactions.

You don't need to worry about increasing maxconnections count. Default values are just fine, and after a few days will typically lead to a stable number of 20-50 peers connected, assuming your router/firewall has been set up to allow incoming connections on port 8333 to your computer. This can be easily set up from the client if your router is uPnP compliant, or you can manually configure the router to forward port 8333 to your PC. If incoming connections are blocked your client will be limited to eight connected peers.

Let us know how syncing is progressing!