Wavelets and bank filters The wavelet transform has a problem as it gives poor time resolution for low frequencies and poor frequency resolution for high frequencies according to uncertainty conditions.

This appears well while using the window notation. But when using bank filters, I can't imagine this problem. So, do bank filters solve this problem?

In addition, if there is a signal with maximum frequency equal to 1000 hz, how are low and high pass filters designed to decompose the signal according to a certain mother wavelet?
Filter banks are one way implement specific discretized versions of a given continuous wavelet transform (or a given SFTF). Their performance depends a lot on how one designs wavelet voices and their subsampling. 

if you to the dyadic version (or DWT), then the highest subband spans a $[1/2,1]$ (in reduced frequencies), the next one $[1/4,1/2]$ and so on. So the  frequency resolution is usually poor on the highest subband. Alternatives consist in further dividing each subband (wavelet packets), or in already splitting the data in $M$ subbands instead of $2$ ($M$-band wavelets), or a combination of the two.

Last, the filter banks are quite generic, so nothing prevents your to use an augmented one, with both wavelet-filters and STFT-filers.