Question 1) You can make the checks when he goes back, or he can make the checks himself every day. The business works even if the owner isnt there to activelly direct the workers, as it is assumed you hired people to make your business work, and your earnings are your profit after you paid them and the business paid all it's cost.
Though, if the hero isnt there, you should apply the 7 days absence penalties as normal, which could ruin the business after a while (which can be fixed with a manager).
Question 2) The manager's job is to prevent the absence penalties and roll for the random events instead of the hero, but the earning checks will happen wether or not the hero is there. Personally, i prefer to roll them after they get back, we count the days away, i roll a couple of random events, see how many checks they have to make, and they take 10 on all of them.
Question 3) As said above, the checks will happen regardless of his presence. But that is a good question because if the hero can communicate at distance, the absence clock is reset everytime he communicates with them. That way they know what to do and they know their boss is still alive.
Question 4) Everything other than GP you will have to pay the half-price for it.
If i recall correctly, labor and goods are 20 gp (half price is 10 gp), while influence is 30 gp (half price is 15 gp), and magic is 100 gp (half price is 50 gp).
So, when you are earning capital you dont gain any GP, but you are not actually losing any either, because the price for rooms and hirelings are based on the full cost of the capital, so if you take your time to earn the capital, you can buy things at half the cost.
See that as an investment of your money, in the long run you will have discounts to build other rooms and hire more people.
There's no real contradiction here. In general, anyone can use a tool with or without proficiency - if they have proficiency, they get to add their proficiency bonus to checks made with that tool. For example, from artisan's tools:
Proficiency with a set of artisan’s tools lets you add your proficiency bonus to any ability checks you make using the tools in your craft.
However, as you've already discovered, crafting has specific rules:
You must be proficient with tools related to the object you are trying to create (typically artisan’s tools).
So in general, proficiency with tools isn't required to use those tools, and Mearls' answer reflects that, but in the specific case of crafting, proficiency is required.
Best Answer
Before committing to this route, ask the GM if the campaign will even accommodate a PC using during play the skill Craft. Many campaigns don't or can't accommodate mundane crafting well at all because mundane crafting takes a long time—a basic sword can take weeks and an expensive, extravagant item made of special materials can take from several months to over a year of constant work. Even a master armorer or weaponsmith often finds it more convenient to purchase his gear instead of making his own. (The Craft skill can be used untrained, though, so it's totally legit to compose a character background saying that a character was, like, a weaponsmith's apprentice before he started adventuring yet the character possesses no ranks in Craft (weapons), quitting his master before he took any ranks. That's a thing.)
Tools and raw materials make crafting happen
Assuming the GM's campaign supports mundane crafting, as other answers mention, at least improvised tools if not artisan's tools or better are needed to practice a craft. These are abstracted sets of generic tools needed by a craftsman to practice the craft. Unless the GM rules otherwise, these are all the tools needed to practice the Craft skill. A separate anvil, for instance, shouldn't be necessary unless hunting roadrunners or needed for a feat. Also, usually tools and weapons are separate things—a warhammer suitable for bashing in orc skulls may not be a good tool for hammering steel.
Keep in mind that each set of artisan's tools is good for but one Craft skill—artisan's tools for the Craft (armor) skill are not also applicable to the Craft (weapons) skill, for example. Luckily, Craft skills are broad—the the Craft (armor) skills covers all armor and shields from studded leather to heavy wooden shields to full plate and the Craft (weapons) skill covers melee weapons from quarterstaffs to flails to greatswords—so the tools are broad, too. To create a quarterstaff a craftsman does not need artisan tools for both Craft (carpentry) and Craft (weapons), for instance.
Finally, a craftsman usually can't make something that costs something from nothing, so a craftsman typically also needs raw materials. These are the abstracted parts that are needed to create the item, and these raw materials usually cost 1/3 the cost of the finished item. For instance, a longsword costs 15 gp, so the raw materials for a longsword cost 5 gp. Ask the GM if a craftsman buys raw materials that are either generic and can be used to craft whatever or broad and can be used to craft anything within a certain skill or specific and raw materials must be purchased for each item the craftsman intends to create. With the looming potential of an accounting headache, pray it's not the last.