Babies that age are much more interested in the human face, expression and interaction, than anything else. We are gregarious creatures by birth, and instinctively seek to socialise. Her favourite game will be you
At this age she might be just getting to the stage of reaching for a toy, but holding and playing with a toy deliberately takes a little more effort and finer motor control which she won't have developed completely for some time yet - they use their hands as a fist at this age, so it is easier to bat out at something, and therefore a selection of shiny or bright coloured objects hanging just at fingertip reach will buy you a moment of peace. They may appear to like the tv, but it is only for the colour and sound, not content, and their attention span is not very long. Toys and tv might buy you enough time to make a cup of tea, or go to the bathroom, then you are expected to take place of the toy.
Re the teething. A baby's teeth move around in the gum for some time before they are actually cutting, it is debated as to whether this causes discomfort or not. My own opinion is that they are just getting to the knowing stage, so their grumpiness can also be a demand for attention, and instinctively every child will put things in their mouth (teething or not) it is how they explore texture, size, shape and taste. Their gums are more sensitive than their fingertips at this age and their little minds are more advanced than we often give them credit for, and they will continue to surprise (and sometimes even humble you lol!) by your underestimation of their knowledge, and search for understanding. It is relentless, until their teenage years
Try laying her on her back on your lap and pull faces at her. Watch her try to copy you and raise her eyebrows, blink, and stick out her tongue. It is quite amusing! At this age, she is also ready to engage in "conversation" She will coo at you, looking at you intently, and when she pauses she is waiting for you to respond. You should also pause and give her time to respond to you. Once you lose the eye contact, she has had enough.
Bear in mind, though, a baby will grizzle if over-stimulated too. We all need some quiet time, and soft music (even better, you singing or reading) will relax her
