1. Doubt it. Could play a small role, but I imagine there are very few places that wouldn't consider someone from a Counseling background if everything else was a fit.
2. Do you have clinical experience in VAs and/or other relevant settings? A pediatric background could potentially be framed as a strength assuming its pediatric psychology in the strictest sense. It would mean you have experience working in medical settings (fuses well with primary care and other VA emphases) and some VAs are now also pushing for more integrated family care. That said, if you have limited clinical experience with traditional VA populations, I imagine it will be a very tough sell at internship. For practicum...just make sure you are getting broad cutting clinical experiences earlier on.
3. Varies by VA - those with strong academic affiliations will have tons of opportunities (e.g. West Haven, Boston, Palo Alto, Durham, etc. - far from exhaustive, just the ones I'm most familiar with). Note that it is usually of a different type than in many other settings and that in general you won't find many powerhouse researchers, giant labs, basic science support, etc. as in a typical AMC. Access to salary support for research varies and the nature of the work is often somewhat different. That said, I think there is more than enough opportunity for what most people want...its just usually not as great a fit for a primary appointment among the hardcore research crowd (though again - there are exceptions).
