No, you can't say this. It's a common mistake among speakers of English as a second language. You can say either: 'I forgot my keys' or 'I left my keys at home'. You can't combine the two sentences.

Yes, if you would like to specify a place you could use "I left my keys at home"

It's a common myth among pedants that "forget" can't be followed by a location. It can. Native speakers use the construction all the time. Search the web and you'll find many forums where some people claim it's wrong, while others (whether from the US, the UK, or elsewhere) say that they use it and commonly hear it. Although people argue about it, there's no grammatical reason why forgetting can't have a place.