This should depend on the goal of your study. If you intend to show that your weighted graph is somehow different from an arbitrary weighted graph (under some restrictions), then Erdos-Renyi (with the same restrictions) would be appropriate. That being said, if you claim that your specific graph is somehow different from others with the same degree sequence, then of course you would need to restrict the degree sequence.

As for randomization if you need to preserve degrees, the first thing that comes to mind would be random edge swaps but then there's a question of mixing time and I would suspect there's a better way to do it. I would start by checking Nick Wormald's work on random regular graphs.

It depends on what you really need as mr Magnant said before. Anyhow, if you need to do a null model that preserves both strengths and degrees, I recommend papers from Diego Garlaschelli (they my put you off if you are not a physicist, but with some help you can read them and use his method). Rewiring weighted networks produces some bad results, because it either distributes weights strangely or it creates denser networks. At the moment as far as I know the only algorithm that will work well is the one by Diego etal. If you are using directed weighted networks then the optimization problem can be really hard and will not work good for large networks. My student did the implementation of the algorithm for his master thesis and the networks had to be smaller then 1000 nodes in order to get results in meaningful time on normal computer.

Would love to but the algorithm is already published by Garlaschelli et al. We needed it for analysis and I did not create it. ATM the best algorithms for this kind of analysis are made in Garlaschelli group, so unfortunatley I can not fish for citations regarding this question :). Nevertheless If I will have something along this lines I will gladly submit. I can ask Diego if he allows and then we can post the code on github if he agrees.