Let's start with some general technical info, and then we'll get to the specific questions.
One way to judge the warmth of a down/synthetic "down" jacket is the type and amount (by weight) of insulation. Type is easy to figure out. Brands will say if it is down or synthetic, and typically the fill-power of the down or the type of synthetic. In terms of amount, some insulated jackets from outdoors oriented brands will just straight give you a fill amount in terms of g/m^2 for synthetic and total g/oz for down on the product page. Anecdotally it seems to have gotten less common in the last 5 years though.
Let's use Feathered Friends Eos as a down example. It says it has 4oz of fill of high quality 900+ fill-power Down. Fill-Power is a numerical measure of expansion/loft of the down (higher number = more loft = warmer by weight). While there can be some exceptions based on shell fabric/construction/fit this jacket will be warmer than a jacket with less insulation, or with lower fill insulation of the same weight. But 9oz of 750+ fill-power down is going to be warmer despite a lower fill-power. To compare jackets it is a good rough estimate to just multiply the two numbers with higher being better.
Let's use Arctery'x Atom LT as a synthetic example. It says it has 60g/m^2 Coreloft insulation. Coreloft is their in-house synthetic insulation. All the different types of synthetic insulation from reputable outdoor brands are relatively similar in warmth. The quantity is the key number to look at, and, exactly as you'd probably imagine, higher is warmer. Since synthetic insulation is almost always manufactured as big sheets that are cut to shape for jackets you'll see the measurements as g/m^2.
Why do they measure down/synthetic fill differently? Primarily because of the practical manufacturing difference. One is a loose insulation whereas the other is flat sheets. But this doesn't mean you can't compare. Anecdotally, I would say jackets with 3-4oz of 800-900FP down are about as warm as 100g/m^2 jackets. My Cerium (3-4oz of 850) is good for 15-20F degrees more low-activity warmth than my Atom LT (60g/m^2).
A less scientific way to judge warmth is price. You get what you pay for mostly rings true with insulation. It's of course non-linear. But a $100 down jacket will not be as warm as a $300 down jacket. Or at least I haven't found one yet.
Can you judge a jacket by how warm it is when you first put it on? Or do you need to move around first to build up the heat?
It's very hard to judge a jacket by how "warm" it is when you first put it on because it's a very unscientific measurement. A jacket's role is just to provide a thermal barrier. It doesn't create warmth. It prevents body heat from escaping and "keeps out" cold air (Technically it's all heat traveling at a thermodynamics level). So the warmer you and the jacket start, the warmer a jacket will feel when you first put it on.
Is it possible to have a jacket that is so warm that you don't need to
exert yourself first, but you can still be warm standing/sitting in
the cold?
Your body is always generating heat. When you're active you just generate more heat than when you are still. So, yes, it is possible to have a warm enough jacket that you can be still and still stay warm. Many companies make synthetic "belay parkas" that are designed for exactly this while belaying climbers. They are 150-200g/m^2 insulated jackets that are oversized to fit over top of other layers while standing still on cold belays. Most people have no such use for these jackets.
So what jacket should you get? For standing around in 0C I'd actually recommend something around the warmth of the Patagonia down sweater you currently own. My Cerium LT with a similar amount/fill-power down keeps me happy at those temps. If you are still cold you could look at something with more fill, or just throw on a layer underneath.
