How do you do spells from the standpoint of memorization? Do clerics and druids memorize, or is it just the wizards, or nobody at all? All spells need to be chosen at the beginning of the day. Clerics/druids pray for them. Wizard study.
What age category are the dragons? very young and young
Would druid characters have an additional animal companion or would the dragon be the companion, or either? druids would not have an animal companion - the bond with the dragon would prevent it. Similarly, magic users would not have a familiar. Also, druids will not be able to shape shift (when they reach that level) as they normally would - a different ability will be substituted. :)
Are you thinking that this campaign will be mostly outside and in the wild or a solid mix of city and wild? I'd say a fairly solid mix.
Are there huge cities in this world where high-level magics and such can easily be found? Yes. But not near your current home base.
Do we get starting gold for a 3rd level character? No, but you may assume basic adventuring equipment plus 10 gold (so, pack, blanket, rope (not silk), etc.)
Do we keep our dragons with us 24/7 or do they tend to stay in their respective homes? You have been with your dragon companion since it hatched.
Should we treat them as steeds, wards, companions, masters, or something else entirely? Companions, soulmates, protectors. Death of the guardian (that's you) is traumatic and unpleasant for the dragon, but not fatal. The reverse is not true - if your dragon dies, so do you. So your relationship should reflect this.