Welcome to the Forums, Csaba! You are correct in your analysis, except that the first sentence could mean that the doctor intended to share your bedroom hospitality. It can be solved with: The doctor suggested that I go to bed.
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GPYNormally they would be understood to mean the same, namely that you should go to bed (alone) because of illness. The second version with "my going" seems more formally phrased.The other interpretation is possible, but it needs a context to direct the reader's or listener's thoughts in that direction.You're right. The second version is excessively formal.
CsabaAs far as I'm concerned the first sentence means that the doctor suggested that I should go to bed with him/herNo native English speaker would interpret it that way, unless you and the doctor were in a bar or other social setting and were physically attracted to each other.