Having your components run at a lower temperature has virtually no relation to running them with an inadequate power supply. Running at higher fan speeds will reduce temperatures, but it won't make your components accept the poor PSU and adjust with it in the short-term. They will still consume the same power they need for working at rated speeds, and in fact the higher fan speed will probably put a slightly greater load on your already ill-equipped PSU.
No, higher fan speeds will NOT make your PSU handle more load than it is capable of handling. Running fans at higher speeds is possible, but the power used by the CPU and GPU will not change due to this, and the load on the PSU will be the same/higher due to faster spinning fans. Long story short, you need to upgrade that PSU regardless of what fan speed you run the PC at, and upgrade it as soon as possible.
Doubt anything will happen to the fans if they run at 100% for a few hours, but using that PSU is a risk for every minute of the day. Only life span you are risking is the CPU and GPU itself, because that PSU could kill both of them any instant under load. The fans have pretty long life, so using them at 100% for some time might reduce their life span, but it probably won't be a very significant reduction.
It'll help a little bit, but honestly it's still a risk you should not be willing to take. Still, I know what it's like to be limited by circumstances when it comes to buying PC parts, and if you are willing to take the risk for the short-term, then you can go ahead and use that PSU until you get a replacement. Limiting the power usage on the GPU might help a little bit, but still try to make sure you don't push the PC to its limits too much or too often.
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