William Burrows, in his book Richthofen - A True Story of the Red Baron, claims that rumours of a romance between the Rittmeister and his nurse, Katie Otersdorf, were untrue. It was said at the time that the couple had an intimate relationship and she sent him scented letters upon his return to the front. The letters have never been found. Burrows also states that there was gossip that they had a child together, a son, who grew up never knowing his father's name. All grist to the Hollywood mill I suppose. However, it has also been suggested, albeit obliquely, that he was a homosexual. One friend of von Richthofen, Leutnant Hans Georg von der Osten, claimed that although von Richthofen treated all women with profound respect, he was 'indifferent to them romantically and never had a sweetheart."
I have a copy of an Over The Front
article (Vol 12 no 3 "The Girl he left Behind", Jan Hayslett and Suzanne Fischer), which also suggests that the Katie Obersdorf story is purely speculative. They do suggest two other names. There was an Austrian girl called Polldi "Lea" Schwarz, whose name was linked to MvR and there are suggestions of a liaison. But again, little real evidence: there is one story that she visited the Richthofen family home after his death and was turned away, and the likelihood is that being of a lower social class than he, an engagement would have been very unlikely.
A slightly better candidate is Baroness Adele von Wallenberg Pallachy, a year younger than MvR. The authors seem to accept that they became engaged about six weeks before his death; she never admitted to this in later life but some of her family appear to have confirmed that this was the case.
As to whether he was homosexual: this is speculation deriving from the anachronistic perceptions of our own culture. He was reserved and not at ease with women in social situations, so he must have been gay, supposedly. I would want a lot more evidence than that. Having said that, I did once see an article that suggested than Lothar von Richthfen and Eric Lowenhardt were more than good friends. But even there, we would need to have a lot more knowledge about how to intrepret the culture of the time to be able to interpret the evidence. Not that it matters, either way...