I settled in on Schwab brokerage about 15 years ago, so since most of your quetsions are brokerage specific questions, my answer refers to Schwab. My guess is any competent brokerage house would provide these options.
Tracking investments: Most brokerage accounts tell you exactly your lots for each investment (if you have multiple investments in the same thing), so you don't have to track anything. They do. They list your cost basis, current market, and gain or loss for each security holding. I really like it now because I can click % at the top, and it ranks it based on the highest % return, and that is typically what I use when I harvest select. In the old days, it was a nightmare. I remember the binders I had to keep when I was starting out investing $50 a month in four or five funds...It is easy today.
Directing the sale - I set my default "sell" option as lowest cost - that way when I select to donate, it uses that and picks the lowest cost lots from each security I pick to donate. In other words, if you have 5 shares at $5 cost basis and 5 shares at $6 cost basis and tell them to donate 6 shares, they would take the 5 at $5 and the 1 share at $6. You can select lots when you sell, but in that case you might want to pick the highest cost shares so you can create a capital gain loss or minimize the taxable income.
Timing - When you are doing DIK's it is more about convenience. When I have lots of high gains sitting in my account and I think the market is high, I tend to donate in January for the tax year - you never go broke taking a profit. However, the last few years, that cost me a lot of money, because it would have gained more if I donated later in the year. In most years, I tended to donate mid year, thinking 1/2 prepaid, 1/2 in arrears. It has been a long time since I donated in December - I prefer to get it done so I don't have to worry about it. Plus the market seems to get volatile the last few weeks of December...