Here is a link to the service manual for your machine. I have a 211U566 which in similar to yours. http://parts.singerco.com/IPsvcManuals/221G155_211G156.pdf. Look this over carefully. I see no reason your machine won't sew the material in your picture. I sew 3/8 " hard leather from time to time with this machine using standard 135-16 needles.
Your machine came set to have 1/2" presser lift with the knee lifter. If not adjust it to spec. Ensure your needle is not bent. You should have little to no play when you wiggle the bobbin case in the hook assemble. Check by putting your finger on the bobbin latch and wiggle. If there is a lot of play it can cause some of the problems, but becomes more pronounced when shifting to reverse which your machine does not have. You should also pull on the thread with the presser feet down, but not threaded through the needle. Using low thread tension you should be able to pull the thread by hand. If the thread acts like it wants to break using say 92 thread. Stop you have a tension problem and you may need to replace your tension assembly. They are less than $30.00. Remove your bobbin case (remove three tiny screws) and inspect where the thread exits the bobbin case and ensure no burrs and that no thread is wrapped around the hook shaft, Also inspect the bobbin tension spring these seldom need adjusting unless you are running very large thread. You mention you are using a needle 3/16" longer than standard. Your machine has two engraved lines on the needle bar. When your machine has the needle in the lowest position you should see these two scribed line. They are used to set the needle bar height when using the standard 135x16,17 needle system. They will not work for the needle you are using and you will need to scribe new ones should you choose to continue to use your current needle which I see no need for in the projects in you photos. The rise from lowest position to where the timing is set is 3/32" for the 135X16 needle system. About 1/16" for the 134X35 system and I'm not sure what it is for the system you are using. One other thing to check which also can cause thread damage as you show, is the hook height. This determines the clearance between the bobbin case stop and the throat plate. The spec is .032" (0.8mm), if it is too much less the thread will bind when trying to pass between the bobbin case stop and throat plate and shread thread. You will also have your situation if the hook guard is not set to keep the needle from bending and the hook point from catching the thread. One other thing to watch out for. Your machine can handle a 135x17 needle up to number 26. Very few presser feet ,however, will allow a needle that size to pass through some won't even allow a number 23 needle to pass through let alone the thread and needle. Make certain the hole in you presser foot is large enough for the needle and thread you are using. But before doing anything change or needle.
Regards, Grant