Firstly, adherent cells when seeded at an appropriate seeding density in the flask, will adhere to the substratum within 24 hours and start to proliferate under the right culture conditions. These can be regarded as healthy cells. However, when cells are stressed, either the adherence to the substratum is delayed or the cells fail to attach. This could be due to many reasons such as faulty freezing or thawing techniques followed, contamination, inappropriate culture media (poor quality FBS), temperature fluctuation in the incubator, etc.
Secondly, during subculturing, if trypsinization is carried out on an over confluent culture, cells may die due to lack of nutrients and space resulting in the release of DNA from dead cells into the surrounding. The DNA being sticky in nature, will form clumps of cells and debris when one trypsinizes an over confluent culture. Also, trypsinization of such over confluent culture will result in detachment of cells in the form of sheet of cells.
So, one should trypsinize cells when the culture is 80% confluent so that one obtains single cell suspension which are in good condition ready to be sub-cultured to the next passage.
To complete the answers given above, you can also look at the morphology and granulosity of your cells.
Both HEK293 and HepG2 cells tend to grow from small islets of usually triangle-shaped cells, and you can almost always observe a few small black dots in the cell itself. But if you observe more, especially in the periphery of your cells, combined with slow growth, no attachment to the culturing surface, and a bizarre morphology (I'm sure you can find nice pictures of HEK293 and HepG2 normal morphology in the literature), it is a sign that your cells are not happy / healthy.
Another sign of unhealthy or suffering cells is the appearance of vacuoles in the cells : round white "bubbles" that are clearly visible, and that are not there when your cells are happy.
Also, your medium is a good indicator of your cells health : if you can observe a lot of cells (especially round, granulous and black- or grey-ish cells) and/or cellular debris in the supernatant, especially in the first 24h hours after your passage, it usually means that something happened. Too much trypsin, that stayed too long on the cells ...
The dilution you apply to your cell culture is also important : if you dilute too much your cells, they might have some difficulty to recover and to reach confluence again. Where I work, we usually try to reach confluence before passaging the cells, but HEK293 and HepG2 are not really "difficult" cells and they don't mind a passage at 80%, even less. Also, for some experiments the confluence needs to be adapted and not reach 100%.
I would not suggest to change the medium too often, as the cells exchange molecules that are released in the culture supernatant : once every 2 days, if your cells have not reached the right confluence, is enough in my experience. The only exception is after the thawing, as you might have more dead cells than after a simple passage (freezing-thawing is a stress for your cells, even when performed in good conditions).
Hope this helps for your cultures and experiments !