Answer to "The Watch won't exit safe mode."When you do a reboot by holding the side button in, at the LG screen swipe from 11 to 3 and when the fast boot menu comes up press the 0 at the bottom of the screen for a factory reset.
Answer to "My watch is not charging at all"You may need to change the battery, see other answers but try a factory reset, while keeping it on the charger press and hold the side button for a minute or so until you see the LG logo then let go, it may then do a reboot. If not - when you get the LG logo swipe from 11 to 3 and you may get a 'bios screen' at that point press the 0 symbol at the bottom of the screen. Odds on it will require a replacement battery I think.
Answer to "Where can I buy replacement parts for my lg g watch"http://www.fonejoy.com/genuine-lg-w150-urbane-watch-bl-s3-410mah-battery-eac62618501.html This is a link for a replacement battery that works and with carriage it's about £21.
Answer to "Watch only works on the charger"http://www.fonejoy.com/genuine-lg-w150-urbane-watch-bl-s3-410mah-battery-eac62618501.html Had the same problem and yes I bought a new cradle too. The link supplies a replacement battery with carriage for about £21. You need Torx 4 bit and follow the tear down. Getting the band off was the worst bit, go very carefully with taking the connectors off but otherwise Ok. Just done mine and is now charging up fine. On arrival the new battery had 30% charge. Recommend the Ifix tool kit for jobs like this! PS don't worry whether it's an LG Watch R or an Urbane they both use the same battery.