I don't like spoiler tagging speculation, but as the OP has I will continue a like.

I wonder if the space suit is also used to prevent the paradox of older River and younger River being present at the same point of time tearing the Universe apart

Could be to prevent some sort of paradox. But younger and older River being present at the same time wouldn't necessarily create one. We've seen it happen before. If you are thinking about Rose and baby Rose in Father's Day, the only reason a paradox could be a problem was because Reapers were present as Rose had stopped Pete from being killed.

Older River was in the spacesuit in 2011, and the little girl was in it in 1969. I suppose the Silents were developing it in 1969 and keeping her in the orphanage, but they were disrupted by the Doctor's actions later in the two-parter so somehow they lost Melody, and she wanders off by herself before dying and regenerating. I'm a bit confused by how much control she had over herself after that - they must have gotten to her again, as by the time we saw her as Mels she was pretty intent on killing the Doctor, but when she was the little girl she was terrified of the spacesuit and begged for help to get away.

I think we have to assume they did because how could Mel's be a toddler in 1969 and then a pre-teen in the mid 1990s as a friend to Amy and Rory without subsequent regenerations? I guess The Silents must have taken her to the future. Either that or as part Time Lord she ages at a slower rate during each incarnation. The Doctor aged hundreds of years in his first incarnation, but looked like a man in his late 60s, early 70s in human terms. The only problem with my second suggestion is that Mel's seems to have aged at a human rate from the 1990's to 2011.