Capital accounts are credit accounts. That means they are increased by credits. And they are stated on the Summary from the perspective of the business. A positive balance means the business owes money to the owner of the capital account. That is, the owner has contributed more capital than has been withdrawn. A negative balance means the owner has withdrawn more capital than contributed, normally because of profits taken out of the business. It sounds like you already knew that. But I reiterated these basic principles as context for my explanation.
When you purchase something for the business with personal funds, an expense claim debits an expense account and credits a capital account (assuming the expense claims payer is a capital account owner). Therefore, the capital account moves in the positive direction (becomes less negative). So, your statement that an expense claim allocated to the capital account should increase the balance is correct. But increasing the balance means moving it in a positive direction, not making the negative balance more negative. Your personal funding of a company expense is an equivalent contribution of capital. The business now owes you more money.
At the same time, profits distributed from Retained earnings debit that account and also credit the capital account, making it more positive. This is because the business no longer retains those profits but now owes them to the capital account owner.
So, both expense claims and earnings distributions are having the correct effect. Your second question actually highlights the source of your confusion. If you drill down on the capital account balance, you do see the number with a reversed sign. This is simply the result of traditional accounting conventions. Balance sheets have historically shown all balances as positive numbers when they are in their expected range. So asset accounts show as positive numbers, but so do liability and equity accounts, even though they really have opposite mathematical signs. (The exception, of course, is contra accounts, which normally show as negative balances.)
If you continue drilling down on capital accounts, you eventually get to a screen where transactions are shown as debits and credits rather than positives and negatives. These are what really matter. In fact, the choice of positives and negatives for debits and credits is an arbitrary one made by the programmer. You can build accounting software where the choice is reversed, and everything will be just as accurate.