The easiest route is to package everything in a Jar file. The default screen does show a progress bar with some small ability to customise. You can write custom code to keep track, manage and download files but I would personally advise against this route. If you search for applet loaders you will find more information.
I'm not 100% sure what you mean, but you can use Swing components in the same way you can use them in applications. Use a JButton with an image is quite trivial, then hook the event code in the actionPerformed method.
The biggest problem you will probably come across is animation and the EDT. I asked about this earlier.
This page has a whole bunch of useful links for game development. Pulp core is an open source framework worth checking out - even if you don't use the framework you can investigate the code.
Whether you should use Java applets or not seems out of scope of this question, but a lot of the above answers give objective (or no reasons at all) about whether to use Java applets. If it's a game for a personal exercise to learn Java then it's a great approach. If you wish to make it public you need to consider whether the current adoption levels are high enough for your needs.
Things have changed in the applet world recently. Since 1.6 update 10 it is much more competitive with Flash - the download size is smaller (at typically under 4Mb), the startup time is reduced and a new scaling look and feel was introduced.
