Yes. In case of mixed gender you should use "wasze (dzieci)", "wasi" are only for pure-male group.
This will also appear in numerals. Only males - ending "ch (ends with 2-3-4)/u (ends with 0-1-5-6-7-8-9 AND 12-13-14)" - "dwóch, trzech, pięciu, dziesięciu". Only females without changing of form - "dwie, trzy, cztery, sześć, dziesięć, czternaście". Mixed group/neutral "oje/oro" - "dwoje, troje, czworo, sześcioro, dziewięcioro"
Not true. Wasi (the personal masculine form) is for groups with at least one male, eg. wasi rodzice (your parents), wasi przodkowie (your ancestry – both male and female) etc. Typically when you refer to mixed group of males and females, you use masculine personal plural.
The children are an exception here, because Polish language treats them as neuter, and as such not as male persons, so a group of children (dzieci), even with mixed genders, in plural is treated by Polish grammar the same way as siostry (sisters, all females) or koty (cats, not personal): wasze dzieci, wasze siostry, wasze koty.
On the other hand, chłopiec is masculine personal noun, so wasi chłopcy.
You are right about numerals: they are different here, because they use those -oje/-oro forms for all mixed groups (dwoje rodziców, dziewięcioro przodków, dwoje małżonków), as you wrote above. Male forms for all-feminine groups (but this also accounts for inanimate feminine words like dwie książki – two books) and masculine personal forms for only male persons. There are another forms (dwa, trzy, cztery… the normal forms) for not personal masculine nouns, like dwa psy, trzy kamienie… Polish numerals are a mess. :P
Where can I find more help online for numbers? The are my Achilles heel when it comes to polish.
If two nouns in my possession share the same gender. Is it possible to skip the possessive pronoun for the second noun?
Mój pies i kot.
Or must I still add it regardless?
Mój pies i mój kot.
The answer is... I don't really know. Two course moderators said that it seems fine here. But I also asked on a Facebook group dealing with correct Polish language, and all the comments said: you should use plural. Now, I may have not chosen the best example, because I asked about "my brother and my cousin". They argued that I have to say "moi brat i kuzyn" or otherwise it seems like they are one person (jokes about rednecks, anyone?)... Now, of course my dog and my cat aren't the same thing. But I really don't know if your version would be correct... sorry :(
BTW, to me this 'moi' thing doesn't sound well... I guess that actually this is not a construction that you use often, for example you'd be more likely to say "moje zwierzęta" instead of dealing with the dog and the cat separately. But it sure is an interesting problem.
"wasi" sounds something like "waśi" here. Is it the voicing that is wrong, or does the i palatalise the s, or something else?
It's pronounced really well here. si is indeed palatalized, very similar to ś itself, but you gotta pronounce the i sound as well, so let's say that it's kind of like śi. So will it be for ni (ńi), ci (ći) and zi (źi).