What combinations of alleles could result from a crossover between BD and bd chromosomes?
The combinations would be BD, Bd, bD, and bd. A better way to write the initial genotype would be to write it like this: BbDd. Writing it that way shows that the organism is hybrid for both traits. The way that you wrote it does okay, but it could cause a person to think that two different individuals are being examined. Writing BbDd clearly shows a single individual that is heterozygous for two different genetic traits.
Alleles will sort themselves independently, so possible gametes will have the "B" trait and the "D" trait in them. Your initial genotype leads to four different gametes, because the genotype is heterozygous for both traits. One gamete could be BD. That gamete is dominant for both traits. Two possible gametes would be dominant for a single trait. They are Bd and bD. The last possible gamete is recessive for both traits. It is bd.