There is no apparent structure in the plots that you show.
The lag order of those negative partial autocorrelations that lie outside the bands are not multiple of each other (they are lags, 22, 56, 62, 78, 94) i.e., they do not arise after a regular number of lags as for example 12, 24, 36, 48, so I wouldn't infer any pattern based on that from the plot.
As a complement you may apply a runs test, which is a test for independence that may be useful to capture runs of positive or negative values, which would suggest some pattern in the data.
As regards the significance of some of the autorrelations I see that they arise at large orders. You should think if those autocorrelations make sense or may be expected in the context of your data. Is it sensible to expect that the value observed 56 observations ago will affect the current observation? If we had quarterly data, it would be worth inspecting significant correlation at lags 8 and 12 because they are multiples of the periodicity of the data and may reflect some seasonal pattern that we could explain in the context of the data. But I wouldn't concern that much if significant lags arose at lags 9, 11 or much higher lags for which I didn't have an explanation that will justify it as a regular pattern.
As concerns your first question, I would say that there is something wrong, unless you have billions of data points and the interval between each data point is very small. In any case, autocorrelation in residuals is generally synonym of misspecification in the model.
As for your second question, I would say that 'No', you cannot ignore lags that are just above the significance level. Otherwise, that would not be a significance level...
When ignoring important lags, you acknowledge that the data generating process that you use is statistically different from the true data generating process. Therefore, you introduce misspecification into your model.
Whether you are fine with that or not is completely up to you.
Best Answer
These plots look pretty decent to me. I would not expect better behavior even if the model happened to coincide with the true DGP. (You could simulate from the estimated model, fit the model on the simulated data and inspect its residuals to see for yourself.)
Take a look at cross correlations for lag$\neq 0$, too. Given a statistically adequate model, most of them should also be insignificant.