Off-topic until the PC version comes out Monster Hunter World questions below...
1) Early on in the game the answer is no. You'll collect tons of materials while just playing so this isn't a big issue, and this is something I wouldn't worry about while progressing through the story quests. Each monster has their own set of armor to craft with varying bonuses and damage weakness/protection. The story and low rank stuff basically serves as a giant tutorial until you reach high rank (after about 45-50 hours in the game). Once you reach high rank you'll want to plan out an armor and weapon build depending on the different HR and tempered monsters you'll face.
2) Yes, it's definitely worth upgrading weapons and armor while playing through the story. My advice is to try all the weapons out at the training ground and use the weapon that seems to be the most fun to use. The great part about MHW is there isn't a best or worst weapon to hunt monsters. Also, you can't really buy better new equipment. Crafting and upgrading current equipment is really the only way to go.
3) Yes there is a free roam in all of the areas you'll visit during the story. The game breaks down into doing Assigned Quests (story), Optional Quests (story/challenges), Investigation Quests (challenges/rewards), Events (weekly rotation challenges), and Expeditions (free roam). To access Expeditions you just open your map, select a location, and go. It's free roam in a sense there is no timer and you can stay in that area for however long you'd like. Collect, kill monsters that appear, and farm the shit out of any materials you need that you're low on.
4) No. You can fail Assigned and Optional Quests but immediately restart them again. Investigations are quests that you automatically pickup just by killing monsters throughout the game. These have a limited number of attempts before they disappear. They'll reward you with bronze, silver, and/or gold rewards upon completion. You'll get a ton of these just by playing and each investigation usually has between 5-8 attempts. Every attempt (fail or complete) will reduce the number by one attempt. It's not something to worry about though.
I'm a Monster Hunter novice but I've put in 86 hours since release on console (PS4 Pro) and it's been a lot of fun. I might pick it up next Fall on PC depending how good the port will be and what other new releases grab my attention. Since the Fall release schedule is always a little messy and with the Spring/Summer being somewhat bland, I gave into my inner demons and picked the console route on this one. No regrets so far.