1) There is no difference in meaning between быть + instrumental and быть + nominative. Instrumental is more common though.
"У меня была собака" is completely different matter. Here "собака"is a subject and requires nominative only. Literally "A dog was at me." You was misguided by the free word order in Russian.
2) What you describe is called "толковый словарь", you can try to google. However good Language1 - Language2 dictionaries also contain different meanings and examples of usage. There is good one Онлайн-словари Slovoed Online: англо-русский, испанско-русский, итальянско-русский, немецко-русский, французско-русский онлайн перевод, but it seems it is English-Russian only.
3) Причастия (participles) are very common in Russian as well as in English and IMHO there is not much difference in usage. What is the problem with them exactly?