1) I find that the K-9 email client available in the Amazon Appstore is the best gmail client. Works like a charm.
2) Several browser options available if you don't like Silk. Maxthon has one available in the Amazon Appstore. Others are available as sideloads from sources like "Getjar." For battery monitoring, try the Badass battery monitoring software. It will help you diagnose where your battery drain is going. You might also try the "Android Assistant." As far as keyboards are concerned, you're stuck with the stock keyboard. (Rooting may enable different keyboards but I doubt it.)
3) The error message sounds like a website problem possibly combined with incompatibility with the Silk browser. Try the Maxthon browser and see if you run into the same problem.
4) Enable flash in the settings for the web browser.
5) Lots of other useful information is available. Hang around and ask questions.
6) Yes, they work well together. You'll find the touch is a much better e-reader in the sun while the KF has other advantages. If the two devices are registered to the same Amazon account, your progress on one device will be (optionally) reflected on the other device.