The process is the same as for personal cards. No hard inquiry; they will likely just use the last pull they made to give you your Experian FICO 8 score for the decision.
No reason not to ask for 3x, especially if you find yourself bumping up against the limit, unless you also have personal accounts that you may later want bumped up and fear you will hit an internal limit with as you cannot move limits from business to personal (although the reverse is possible). They will counteroffer if they cannot grant the full amount.
If anything the heavy use would just help your cause. Since it doesn't report to your personal reports there is no scoring penalty.
ARs usually only happen if they suspect fraud, or if your spending is greatly exceeding what your finances seem to support. 4506-T requests do happen; the amount of exposure varies for triggering that. If this is your only Amex card it would be highly unusual at such a low amount. Those are optional. You can always back out and ask for a lower amount, or just choose not to supply a 4506-T if you are not interested in doing so.
