Q: If I have a card like Flashlight that indicates an activate action  and Investigate - is it considered an activate action, an Investigate action or both? What about when you play cards with a designator like Spectral Razor? Is that a play action or a Fight action? Both? A: Activating an ability with an action designator or playing a card with an action designator counts as both actions. Using a Flashlight, for example, would count as both an activate action and an investigate action.
Q: Would using Flashlight twice while triggering Haste allow you to either Investigate as a free action or use a card activation action? A: Yes, if you used Flashlight twice in a row and then triggered Haste, you could take either an additional activate action, or an additional investigate action (but not one of each).
Limit 1 per investigator.
After you perform the same type of action twice in a row, exhaust Haste: Take another action of that type again (types include activate, engage, evade, fight, investigate, move, play resource, and draw).
Haste requires you to take two of the same type of action in a row, before giving you a bonus of the same type of action. While free actions are amazing, this means that Haste works best if you can use it a number of times to make up for the cost of doing so. (A baseline would be at least 3-5 times in a game, to make up for the card it takes, the play it takes, and the resource cost.) To confirm its usability, I've laid out the listed set of actions, in a vague order of less to more likely to be done in triplicate:
- Engage: You really don't want to repeatedly engage in most cases. It requires you to have 3+ monsters in one space on other characters, and you to take multiple Attacks of Opportunities.
- Evade: You'll only make multiple evade checks if you're either evading multiple things, or have managed to botch numerous evade attempts. Neither are great situations to be in, but they can happen.
- Move: Three moves is an awkward amount, since it's the equilvalent of spending your entire turn moving. If you really need to get somewhere, it can help, but since it requires you to move three or more spaces in one turn, it's generally better to avoid being in those positions if possible.
- Resource: While it's not ideal, there are turns where you need two more resources to get something. A third resource gain can at least help give you a bit of a boost for the future.
- Draw: Similar to resource, while it's not ideal, if you're desperately searching for a card or need to refill your hand, an extra card won't hurt unless you're pushing yourself over the hand limit.
- Play: If you have a turn where you really need to get a lot of cards out, this can give you a boost. You'll need to either have a good initial hand with Haste included or be in the right place for a couple of events along with assets lined up, but the saved setup time can be worthwhile.
- Fight: Enemies can often require multiple hits before going down. This can give you a buffer if you're using some of the rogue's Opportunistic weapons, or save you a few charges/shots if you're using an Enchanted Blade or a Firearm.
- Investigate: Now we're getting somewhere. If you're not playing Solo, you'll often need multiple investigate actions to completely clear a space of its clues. Even if you are playing solo, this effectively gives you a bit of wiggle room to deal with a failure along the way.
- Activate: This is the most build-dependent use of Haste, but also what likely determines whether or not it's a good addition to your deck. You can activate different items in order to make Haste work, as long as all of them require in Activate action. You can spare a little extra to make an attempt at Burglary or take a swig from Liquid Courage. Go ahead and make a couple of attempts with your weapon of choice before using a Flashlight or Lockpicks to help clue up. Preston Fairmont can mix in an action banking his Family Inheritance. Sefina Rousseau can mix in a pull or two from underneath her card as long as you don't overdo it. Ornate Bow allows for you to work in both a shot and a reload before doing anything else. But if you have the XP to spare, the most flexible mechanism to use is Borrowed Time. In the worst case scenario, you can bank two actions for next turn and get a third free, earning an extra action every round. In other cases, you can make any other activations you have and bank the others.
Either oddly or fittingly enough, some of the more flexible users of this will be people who are already using Leo De Luca, or otherwise have bonus actions to begin with. It becomes much more practical to set up action chains when you have a bonus action to better position yourself before starting, and Tony Morgan can save a bit of ammo when he can take a preemptive couple of swings.
Leo costs 2 more resources but also more than pays off that difference by generating more bonus actions reliably as well as being a 2/2 soak. And it's not like anyone who has access to rogue 2 will have problems with paying 5 upfront.
Baring edge cases like having 3 other allies already, I don't think there's any investigators without Leo who would want to take this. While Dexter Drake certainly likes having more spells, he also has better things to do with his arcane slots.
Does haste also trigger when taking actions on Mythos or scenario cards? What about weaknesses if you have enough actions?
I suppose extra actions make it easier to trigger haste, while free actions that move you also work well to get to more clues.
Edit: Haste makes stuff like "gather ressource" and "draw card" more efficient. Gain 3 cards or 3 ressources for 2 actions? More efficient than Emergency Cache or Preposterous Sketches, if you include costs for drawing, ressources and playing them.
Free actions that grant other action types as effect don't count towards Haste cause you didn't spend an action to initiate them, like Pathfinder doesn't trigger Frozen in Fear. Examples: Double, Double, Uncage the Soul (Sefina), Ever Vigilant (Skids)
Edit2: Ignore my last paragragh, haste cares about performing the same type of action twice, not spending an action to do so. So above cards all trigger haste.