The OE should annex Byzantium, but only Byzantium. The other one-province minors in its turf it should use an annexation ally for. Other major powers should not annex anybody. Doing an annexation is occasionally a good way to start with a one-province minor, i.e., if you start within striking range of Mecklemburg.
As for the OPs questions, you get gold in treasury in just four ways in EU2:
(1) census taxes
(2) tribute, in peace treaties
(3) minting
(4) events
Since you can't control events (for the most part), you can see that the only way to increase treasury is either minting, or wars. (I am assuming you have promoted tax collectors.) In wars, you can:
(a) take new provinces, increasing your census taxes
(b) do synchronized looting
(c) get 100% warscore, then wait around for a few years (while looting), and get the AI to pay you off a lot for peace
Thus, it's very important, when trying to get a fast start, to be able to win wars cheaply in terms of not losing lots of expensive troops. If you lose 10000 cavalry in a war, you'd better be getting a nice load of loot from it or it was not worthwhile. If you can prevail in a war losing only 2000 cavalry, then you can make that up pretty easily in reparations.