Fate has a much more narrative approach, less GM authority, and player-based plot control mechanisms. This can make for trouble transitioning from a more adversarial GMing environment.
Fate Core (and other recent Fate games such as Dresden) actually do a pretty good job of providing a suggested "menu" of powers and stunts for players to take; show them to your players and let those inform other stunt ideas through play, rather than trying to get the characters completely nailed down in advance the way you would have to in D&D.
To address the specific problems you mentioned:
The "blind sniper" problem is that in Fate, it's very easy for a cooperative party to stack multiple temporary aspect bonuses on a situation, then tag them all for a single super-successful roll regardless of innate skill.
Consider a sniper with a base skill of +0 - "has no idea what he's doing". First he hides on high ground (creating aspect "On a Grassy Knoll"). One of the other players - a tech - has made, and gives him some "Precision Armour Piercing Ammo", with suitable aspect. Then another player jumps into the road to stall the target into "Standing Still for a minute", so the sniper can create advantage by taking an acting to put him "Centred In My Sights".
Then the sniper free-tags all of those, for a base roll of +8 and a near-certain hit - with fate points to spare if he needs them.
Stacking enough of this sort of this can greatly reduce plausibility, but it's obvious behaviour for a group of D&D players where it's an expected part of the system to need to stack every combat advantage you can generate. I don't, however, feel it's as much of a problem as some Fate players do - this is behaves-as-designed. Fate characters are supposed to be able to beat pretty much any single obstacle if they can generate a convincing narrative. "He's not that good a shot, but it worked because an entire team was helping him take it" is a pretty good narrative to me, and gives the plot and characters room to develop further.
The "constant compels" problem arises when GM and players get caught in a Fate-point loop of constantly compelling aspects to force behaviour or non-actions from each other.
Remind your players, and remember as GM, that the point of compels is to implement narrative development. The Fate point economy will naturally limit how far players can push this, so it's not a problem unless the GM gets sucked into constantly offering Fate for compels. So don't. Compel when it's a narrative or character development to do so, not just because it's possible.
Long combats occur because groups have trouble actually taking out opponents of similar skill levels. This is particularly likely if your group is used to D&D, and trying to "wear them out" by all attacking individually. In Fate it's much more effective to set up a narrative about a couple of major attacks, taking multiple actions to support each other and make those go off well.
The best counter to most of these issues is threefold:
Let them have their moment. Players being able to tear through obstacles like this is not a problem in Fate the way it would be in D&D; it's expected behaviour of the system. Don't make it impossible, but ask what happens next and generate new obstacles. The problem can come more from a D&D-oriented GM feeling he has to make tasks "possible to fail" than from actual issues.
Your enemies are not idle. Don't GM in direct opposition to the player's actions, but instead change the game. Fate uses the same mechanics for social and combat conflicts, and it does so for good reason, expecting them to intermingle. Use it. Enemies will attack on social fronts, run away if they're in trouble, consult allies, adjust schemes, and all the other narrative options that are not often available to an orc in a dungeon.
Remember that in Fate, the player is always an informed participant, even when the character isn't. The gap between player and character knowledge is much bigger in Fate.
Situation: A hostage is tied to a chair. There is a trap which incinerates the room if anyone touches the hostage.
D&D answer: Tell the players nothing. Make hidden Perception checks. If anyone explicitly searches, make Search checks. Lie unless the players succeed at these checks.
Fate answer: Tell the players immediately that there's a trap. Let them make rolls and use aspects freely to see if the characters spot it. If one of the players has a "reckless" or "rescuer" aspect, compel it - offering a fate point if their character runs in carelessly and sets the trap off. Make the players partners in decisions that hurt their characters.
Summary: Negative aspects are easy "push button here" dispensers for Fate points, but spamming that button needlessly is boring at best.
Two things need to be kept in mind: drama, and the Fate point economy. Being stymied or drained of Fate points by the same problem over and over isn't dramatic or interesting, so don't do it. But the Fate points must flow! So here's how I approach compels in my games.
Encourage self-compels
Most of the time, the Fate point economy should take care of itself. If aspects are interesting and natural to compel, players will often do it themselves! Frequently my job is simply noticing that their natural role-play choices deserve Fate points, and handing out the points accordingly.
If players aren't self-compelling a lot it may be because they feel too overwhelmed by the challenges I'm giving them, so they don't want to invite more obstacles. In that case, I need to back off and give them room to make problems for their characters.
More often, however, it's because our aspects weren't written so it's obvious and interesting to compel them. That means we need to revisit aspect-writing and revise our character sheets accordingly, but in the meantime aspects such as Steers like a cow are great filler to keep the Fate point economy afloat.
Look for compels when pools run low
When players start to run low they often cast about for opportunities to self-compel for more points. Obvious weak spots like Limited battery pack are a useful tool in these situations, especially when their character aspects might not be able to provide immediate traction for compels.
If a player’s pool runs low and they aren't self-compelling, it’s time to start looking for ways to feed them more points. Compels are a great way to do this because invoke points don’t go into the player’s pool until the end of the scene while compel points are usable immediately. This is where obvious but not overly dramatic compels, like "your gun jams," can be great.
Compel when the stakes are high
Drama, drama, drama. When nerves are taut and the stakes are high, throw out compels which will really gum up the works. This is a spice, not a protein, though: use it sparingly and its effect will be all the greater. "Your flashlight dies" isn't very dramatic--unless it dies just as you hear a spooky noise while exploring the sewers!
Best Answer
Does a good Aspect say one thing? Or more than one thing?
We see in much Fate material the advice that a good Aspect says more than one thing. OK: sort of. It should say one thing and then say more about that one thing. It shouldn't say many things which aren't directly related to each other.
Many wordy drafts of Aspects contain information which don't need to be part of the Aspect in question, or indeed of any Aspect at all.
It's not about the word-count, it's about the encapsulation of a single major element of the character (or scene, or campaign, or Extra, or Advantage, or whatever other type).
One Aspect should establish one primary fact. And any color which that primary fact needs.
This does indeed happen a lot: While in "draft" progress, a single Aspect can grow to include (as worded) many different facts and lots and lots of words.
What I do when working out Aspects either by myself (for plot, campaign, scene or NPC Aspects) or with my players (for PC or Advantage Aspects, or some of the other baroque types which players can create, depending on what hacks are in play) is: I ask the Aspect's primary owner what three facts in this lengthy list of words are the really important ones. Three. One of them should be a primary fact, and the other two can color it. If one of the three facts doesn't color the other parts, it gets cut.
It's not that you can't have an Aspect include more than one fact. it just can't include utterly unrelated facts.
"Deposed Pirate King". Primary fact: "Pirate". Color: "Deposed", "King". The only further thing I might consider adding to this Aspect might, might might be if the player wants to say how or why or by whom the former Pirate king was deposed. It could fit here: "Former Pirate King, Deposed by My Own Bastard Son".
I work with the player until the Aspect has what she wants. But I do coach her to narrow a single Aspect to the salient facts about that Aspect. Other facts belong on other Aspects.
If the player must have Aspects which establish entirely unrelated facts, those would be different Aspects. "Gladiator" doesn't color "Pirate" in this case. (Maybe there's some other game or character where "Pirate Gladiator" makes sense, but that isn't what this example was about.) Embellish the "Gladiator" part as much as you and the player want, but every word which is added should be about the fact that it's a Gladiator. What kind, where, whose, how strong, whatever, I don't know, but it must be all about the Gladiator.
But:
Not all facts need to be Aspects.
Having too many Aspects, or too many facts captured in Aspects, dilutes all of the Aspects. It winds up that none of them get their due. Ryan Macklin's thoughts on "aspect spamming": http://ryanmacklin.com/2013/11/fate-misconceptions-and-aspect-spamming/
So: The other thing I do with players in this situation is, I ask them: Which parts of their character details don't need to be Aspects. You only get five.
Just because there are details which don't fit into five Aspects doesn't mean they aren't truth. Every player gets up to five Aspects at character generation, and that is the space within which they can put their Compel-bait and their Invoke potential. This is their moment to say what is and is not going to be important to them in play. They can't have it all, so, if they're in the weeds like this, help/make them cut the weeds.
It does not cost them character richness to leave things out of the Aspects. My players all have character history and other facts which are either understood as being part of what a given concisely-worded Aspect is saying-without-saying, or which are facts which aren't even reflected in any of their Aspects. Maybe you don't have to say in the Aspect that he was Pirate King of the Western Spice Continent's Less-Than-Perfectly-Charted Tradeways. Maybe you don't need an Aspect which says that as Pirate King he enjoyed the loyalty of twenty-two captains and their two thousand scummy villains, and which ports-of-call and pirate conquests were the origins of all of these dependable souls.
It's fine to have character details which aren't Aspects: The fact that my bastard son had the same name as the priest who excommunicated me for witchcraft isn't relevant to my deposed-pirate-king Aspect, but the fact that my bastard son deposed me is. The fact that I'm excommunicated or whether I did or did not actually dabble in witchcraft as a younger pirate might be things which I, as a player, don't ultimately want as an Aspect: It's not who-I-am-today. I don't want Compels on it, I don't want to Invoke it, I don't want to use it to Declare a Story Detail related to witchcraft, the Church, or anything else. It's history, it might come out in roleplay, but there's no room for it on my character sheet. That's for things I really want to be mechanically important in play, and all I get is five slots.
It's great when an Aspect does say more than one thing or does establish more than one fact, but, it still needs to say one main, major thing, along with whatever reasonable amount of color or decoration hangs on that thing. Not multiple unrelated things.