I have the Eurobath and love it as it works from when they are little until the stop wanting to sit still for a bath (varies by child obviously!) and it is just plastic, so easy to clean and keep clean and it is also fairly inexpensive. It is big though, for us it was fine because we just stored it in the guest bathtub, upside down, when not in use and used that tub for the baby only, but not everyone has two full baths. As far as safety, it does a good job holding baby, but for any bath you have to be right there the whole time of course which is what really matters as far as safety. And yes, some folks just do baths in the sink. I think this works just fine, but since I had an extra bathtub that could be devoted to baby, I preferred to use that and keep the sink for dishes.
For swings, I personally would not get one ahead of time as not all babies like them (our DD did not). If you find a gently used one or one on sale and don't mind it taking up space, maybe go for it, but I think swings are very dependent on baby and it is much better if you can try one with your baby at a friend's house or something before purchasing. Those linked swings look fine, but it just really depends on what your baby likes.