How does a prospective applicant fail to qualify for RAship, but fit the bill for TAship? I wish to know how the Graduate Admissions Committees for top Universities in the USA decide about awarding Teaching Assistantship to a prospective student. If one had preferred both 'RAship' and 'TAship' in his/her application, does that mean he/she must first be eliminated from the RAship race and then awarded TAship as a consolation?

Also, if being a Research Assistant requires a stellar undergraduate research record, publications and so on, what are the factors which would have been considered for awarding the Teaching Assistantship?

To be very specific, if the student is from a country like India or China, would his English-speaking skills and academic competence be re-reviewed? 

If so, please help me understand how these two factors transform from mere selection criteria to those for awarding Financial Assistance.

The Universities, I received assistant-ship from are among the best in my field of interest in Computer Science. Also, their CS departments are known to be well-funded and generously award their students with assistant-ships and scholarships every year.
In the US, TAships and RAships are generally assigned by the department. There may be a rule about first year students (ours was that 1st year students could not be assigned TAships). You should also not assume that the division is made by a committee. RAships can be grant-related, or funded by generic department funds. The PI would usually make the decision if a student is considered especially appropriate for the project (unless the PI has a lousy department that doesn't give him/her much autonomy). Depending on the level of democratic sentiment, either the chair decides on the allocation, an admissions / funding committee does, or the whole department faculty does. TAships are also subject to university rules, if any, or department policy, about language competence, so it may be impossible to grant an incoming student a TAship if their English isn't certified for oral proficiency.

Generally speaking, an RAship is better than a TAship from the student's perspective, but the department might either have a particular need for a TA (prehaps in a specific area) in that year, or it might be advantageous to have a year's teaching out of the way so that in your second year you can do something else. I would say that you should not over-interpret either outcome, if you checked the "both" box. "Prefer" means prefer.