After replacing the power steering pump, I am having problems with lines

The high-pressure line will be the one with the short rubber line and steel fittings at each end like the first image. Then there is the line from the rack to the cooler. That one has a steel line on the rack end and a hose clamp at the other end. Then the line that comes from the cooler back to the reservoir only uses hose clamps on the two sections of hose. The only line that should be high pressure is the first one. After the fluid comes out of the rack to return to the reservoir it should only have a few pounds of pressure as it returns to an open path. If you have the clamped lines blowing off there is a blockage somewhere or the clamps are not tight.
You mention filter, there is no filter on the stock system, has someone added one to your car? Could you take a picture of the line and filter please? Post it using the add image button. That might help sort out the problem.

That filter is likely plugged. It isn't really needed as long as you have clean fluid so you could just run a longer piece of hose and bypass it. If you want to replace it many of the parts stores sell a universal one or can look up a filter to fit it. That filter is in the return line, as is the cooler. Easy way to tell is to look at the way they are attached, that small crimp clamp shows low pressure.
It looks like you might have tried to bypass the filter and had those lines come apart, that can be a sign that the line you are using is too small or the power steering lines (ones with multiple hose clamps) have swelled. I would probably go to the clamps where those lines attach to the steel lines, use a pair of side cutters or a grinder and remove the clamps and run a new rubber return line, use high pressure line like you can get for hydraulic lines and use clamps rated for fuel injection or if you are good with tools you could expand the crimped ears on the OE clamps and then crimp those back in place.

It could if it's too thick as that would raise the pressure, but it would take some blockage to do that because the return line should only have the residual pressure as the fluid comes out of the rack. Perhaps bypass the cooler as well, it could be something in it that lets air flow, but thicker fluids are blocked enough to be a problem.

The fluid GM normally uses is almost the same weight/thickness as Dexron transmission fluid but with a different additive package and no dye. GM # 12345866 or the aftermarket version is Lubegard Synthetic Power Steering Fluid.

You might want to look at the clamp designs and use a flaring tool to add a rib or two to keep things together. Or buy some of the crimp style clamps and add a new filter and swap the cooler out and use the crimp style clamps to put it back together. Some of the lines are available new as is the cooler.