Requesting raw data from previously published research
This will, as it seems nearly all questions on this site, vary based on field. My answer applies to Epidemiology and medical research only. Your mileage may vary.
It is very common for this to happen in my field. There has been an increasing emphasis on using meta-analysis and systematic reviews to summarize bodies of work, and with those there is almost always a need for some more information, raw data, etc. to come from study authors. It's fairly routine now. They may not say "yes" for a number of reasons. One may be that they're working on their own projects in a similar direction. But there are others - that privacy laws prohibit releasing data to just anyone or their funding dictates similarly is a fairly common one. So be prepared for "no".
As for the authorship question, it likely depends on the extent of your data request. Generally, if you're just asking for a few numbers that go into a reported value, then in all likelihood, an acknowledgement is more than sufficient. If, on the other hand, you're asking for access to the raw data from their 5 year, many thousands of dollars cohort study? You're likely going to have a member of their study team be an author on your paper, and there will be far more extensive approval processes than just "Sure, .csv file is attached."
"If my model (without fits to specific data) is in a pre-print state then should I send a pre-print to the authors I contact? What if the pre-print points out weaknesses in their approach to modeling similar problems?" For the initial contact, I'd suggest it isn't necessary, because what you're really trying to find out is "Is the release possible". At later stages, I would expect to see what you were doing, either in a full explanation, or as a pre-print. As to your "What if?" question - so what if it does? Science is about improving the methods we use.
Many journal formats (and the arXiv) indicate a "corresponding author". That's the person you'll want to contact.
Doing so isn't a everyday occurrence, but it does happen.
In some fields access to the data may be covered by agreements to keep it confidential for a certain period of time (or until first publication). These agreement often specify the answer to the authorship question as well.
This may or may not be helpful in this case, but you may be able to persuade them to publish the data separately, in a dedicated data journal or other data archive. Depending on the field, they may already have done so (some UK funders, for example, are starting to require this as a condition of funding).
This gives you the additional option to give credit by citing the dataset directly as well as relevant papers, and also opens up the possibility that others will use and cite the data as well, bringing more prestige to the original researchers.
Currently, some researchers are open to this type of data publication, while others are not, so your mileage may vary.
If you want to find data to reuse, or to suggest somewhere for them to deposit, there are some lists of archives available to look through, such as the DataCite repository list and Databib.