One thing I don't understand is why at no point was there any attempt to also communicate this way, [using pictograms] aside from Louise scribbling her name on a sheet of paper in English using the latin alphabet.
What makes you so sure there wasn't any effort to communicate through pictograms? Firstly, the ship that Louise and her team were talking to was one of twelve ships that came to Earth. Secondly, Louise's team wasn't the first in the Montana ship - there was another before her which apparently didn't fare so well. That also implies the other eleven teams were trying things before she got involved, which may not have been discussed on screen.
It's entirely possible that pictograms were tried, and failed to procure a response. In fact, given how the aliens perceive time, it's entirely possible that they were waiting for Louise and it didn't matter what anyone tried until she... Arrived. (sorry, terrible pun.)
Abbot was killed as a result of the bomb that went off inside the ship. ("Abbot is in death process.") But the aliens must have known the explosion would happen. So Abbot knew this was how he would die. Yet, he showed up at the appointed time and allowed the bomb to detonate in his face-equivalent. Clearly, the aliens have stopped trying to change the future they see coming - whether they've proven time is immutable or whether they just gave in is never addressed, but the point is they don't try anymore. They saw Louise as the one who would eventually succeed in communicating with them and so they waited for her.
Was this just a crass American stereotype about the Chinese (the Chinese are just paranoid and aggressive)?
How does the idea of the Chinese trying to communicate through games instead of pictograms make them "paranoid and aggressive?" The Chinese didn't become aggressive until the aliens started throwing around the word "weapon" - misleading translation though it may have been. In a real-world alien contact scenario, I'm not so sure America would respond any differently.
Also, at least two classical Chinese games that I can think of include pictograms taken from their alphabet on the pieces (Mah Jongg and Chinese Chess.) So using these games and using their pictographic alphabet are not mutually exclusive.
An add-on to this: was there any explanation of how the alien language allowed its speakers to manipulate time?
No, there wasn't any explanation of how the language "works." Although, I'm not sure "manipulate" is right word. As mentioned above, the aliens don't seem too interested in trying to change what they know is coming, even when it's bad, so being able to see the future might not necessarily mean being able to manipulate it.