Different sets of forward and primer means: assuming I use forward primer from set 1 and reverse primer from set 2. The reason for doing this is to ensure the primers cover all my target genes for amplification.
Yes. You can specify either the forwrd or reverse primer. This is useful for qPCR where you want a primer to span a splice site. I would try to keep it so the Tm difference is less than 3 degrees 2 better. Good luck!
You...can, but it might not be a good idea. Most primer design software applies specific exclusion criteria based on primer/primer annealing: in other words, when suggesting a pair of primers, it's already checked to make sure those two primers won't stick to each other very easily.
When you pick one primer from a pair, and another from a different pair, those checks have not been carried out, so there's a chance your new custom pair might have an unhelpful degree of complementarity.
If all you want from this is just "a band I can cut out of a gel and clone", then this should still be fine: even excessive primer dimer formation is unlikely to completely prevent amplification of the correct target, and you can select only the product you want by virtue of size.
If you want to do quantitative (qPCR) stuff, or cloning without band excision/clean-up, then this is a slightly less robust approach.
Hello! If you are worried about primer dimers check with the oligoanalyzer. I think that most places you would order primers from, here most often it is IDT. The have a tool. Here is the URL for this, https://www.idtdna.com/pages/tools/oligoanalyzer Novagen used to have a nice primer analysis widget but when they were bought they took it down. There are advanced calculations that can be performed with the custom interface to PrimerBLAST. Good luck!