If you want to see how this works in practice, probably the best way is to read what people have written about taking mulligans. That's really the only decision you can make after evaluating the strength of a hand: is this hand strong enough to keep, or is it weak enough that I'd expect an average hand with one less card to be better?
I know you've said you don't just care about mulligans, but this is really the only kind of real-world resource to point you at, and this is definitely a question where you'll benefit from looking at real examples. When you ask "how does one evaluate the 'strength' of a hand?" as a broader question, the answer in practice is "you don't." All you care about is mulliganing. If it's an obvious mulligan, you don't waste any time deciding precisely how bad, you just shuffle it back. If it's an obvious keep, you might be a little excited if it's really good, but you're not going to try to pin it down. The potential range of power is more like awful to reasonably good, not weak to incredibly strong. Generally, the best possible hand just means you have decent odds and you'll get to play a "real" game.
The Level One series has several articles about mulligans, trying to give some generally helpful advice. If you haven't played a lot of Magic, you might want to read some of the rest too, since that background will help give you the context you need to understand what you're really looking for when you draw your seven cards.
There's also the Keep or Mulligan articles on Channel Fireball, which are all about looking at specific hands for specific decks. That might be a little less meaningful if you don't grok the decks, but it'll give you an idea of the kinds of things people consider.
So this is clearly a pretty complex thing; people are writing whole articles not just about hand strength in general but about specific hands for specific decks! Let's come back to your comparison to bridge, and see why it's so different. For several reasons, Magic isn't as easy to generalize about as bridge:
In bridge, your opening hand is all the cards you'll have for the whole game. In Magic, your opening hand is just the first seven cards - even in a very fast game you'll see at least a few more, but you can easily see another seven, or in longer games, another twenty or thirty. So you have to think about your hand in the context of average subsequent draws. You're far from perfect information, and there's an element of chance.
In bridge, every hand is drawn from the same deck of cards, so you can analyze all hands in the same way. In Magic, every deck is different, so the main considerations for one deck in evaluating hands may be completely different from those in another deck.
In bridge, the cards all come with numbers printed on them indicating their precise "power", and there's not many possibilities. In Magic, not only are there not numbers like that on the cards, there couldn't be. Everything's contextual; it's more like a radically expanded version of nontransitive dice, where it's not "A beats B beats C beats A" but rather a large number of cards each of which are better than some others in some contexts, and worse than some others in other contexts.
In bridge, your opponents are also drawing every hand from the same deck of cards. In Magic, each opponent you play can be playing a different deck, which in turn changes the value of the cards in your deck. Something that's amazing against one opponent's deck might do absolutely nothing against another's.
In bridge, there are actually hands that can guarantee victory. In Magic, there's no such thing. Your opponent will always have at least some opportunity to interfere with your plans, and most often there's a lot of interaction and dependence on subsequent draws. So the power level ceiling is much lower - there's no such thing as a "come" hand. (Even in vintage, where you can play some pretty degenerate combo decks, your opponent can be playing free counterspells to mess you up on your first turn of the game.)
All that said, we can certainly try to come up with some very general ideas.
The most important thing is mana. To first approximation, you're looking for the right number of lands. But as soon as your deck has multiple colors (which it most often does), you're also thinking about whether you have the colors you need to cast the spells in your hand and the spells you're hoping to draw. And since nothing is ever perfect, you're also often thinking about how likely it is you'll find the additional lands you need and the colors you're missing in time.
Next, you're looking for cards that will let you do what your deck wants to do, which varies wildly depending on the deck. In a very simple aggro deck, that might just be a selection of the cards that will let you deal damage as quickly as possible. In a more synergistic aggro deck, you'll want cards that work well together, not just all of one thing. In a midrange, you're probably looking for cards that will help you deal with your opponent's early threats as well as cards to help turn the tables on your opponent in the later game. In a control deck, you probably want cards to deal with threats and some spells to create card advantage (e.g. card draw), but you don't want the cards you'll actually use to win, which you can wait to find later. So even within a single deck, there's likely no simple notion of power, and across all types of decks, it's impossible to generalize. You're just looking for a set of cards that you think will play out favorably, in combination with the cards you'll draw over subsequent turns.
On top of all that, as soon as you have some idea what your opponent is doing (i.e. in the second or third game of the match) you'll re-evaluate all your cards based on how well they fare against your opponent's game plan. Are your threats ones that are easy or hard for your opponent to deal with? Are your answers ones that answer your opponent's biggest threats?
The only real things in common across all of this are that you want "enough" mana and cards that give you a good game plan. What that actually means in practice, though, can't really be said in the same way you can about bridge. In some sense, whenever you build a new deck, you create a game, and then within that game you have to figure out all these things about power and synergy that you'll then use to evaluate hands.
For what it's worth, one could try to force the analogy with your points, and come up with ideas that are certainly part of how one evaluates opening hands.
In decks that have a few "favorite" cards, rather than a more even power level across the whole deck, it can indeed be great to see one of those in your opening hand, like your point (1). This is more common in limited, where you might pick up a single copy of a rare bomb, and not draw it in every game. (In constructed formats, anything that amazing you're probably playing four of, so it's not as much of a surprise to see it.)
Similarly, like your point (2), even if your deck doesn't have a giant range of power, it may have some range, and so some hands may include better cards than others. Again this is more common in limited, where your 23rd best card is often not one you're terribly excited about.
In decks that care more about synergy (how cards interact with each other), rather than just individually powerful cards, the synergy of your opening hand is a consideration, analogous to your point (3). It's a lot more complex to evaluate than simply looking for a long suit, though.
It sounds like the closest analogy to positional strength (4) is simply hands that are awful on their own but if you draw the right card will be great. That certainly happens, but those hands can be anywhere from a regretful "it was so close" mulligan to a hopeful keep (you might do nothing the whole game, you might have a blowout win), So it doesn't really tell you much about actual strength.
But as we've seen, none of those are actually simple rules; the power of a card is by no means straightforward to determine, and there's an awful lot of context to take into account that can completely change things from hand to hand and from game to game. Trying to think of it in terms like this is likely not to actually help you much in practice.
