Is measured floating point speed or integer speed any indication of processing speed for projects? If not what to use as a guide?

I have a motherboard and PSU, and can get an I7-7700k and 32 GB memory for about $80. Should I work with this or just put the money into something more current?

It all depends what projects you want to run.
Some projects do a lot of integer calculations so integer performance is key, for others its floating point, and for others it's a combination of both.

An i7-7700k with 32gb ram is going to be a reasonable performer for many projects, so for 80 bucks go for it, use it, find out what projects work well, what projects you enjoy (remember join the project forum and learn from others).

Is measured floating point speed or integer speed any indication of processing speed for projects?

Very roughly.
While they do give an indication of how well a CPU may perform when both are running the same application, if the project makes use of more advanced instructions than the more basic ones that are used to provide those benchmark values, then the performance of that CPU can far surpass the performance indicated by those benchmark values.
eg A project may have 2 applications for processing CPU work- one for compatibility with older CPUs, another that makes use of more recent instruction sets.
2 CPUs may have the same number of cores & threads & benchmark numbers, but if one of the CPUs supports the newer instruction set then it's performance could easily be more than double the older CPU, even though they both produce the same benchmarks.

I have a motherboard and PSU, and can get an I7-7700k and 32 GB memory for about $80. Should I work with this or just put the money into something more current?

That's very much up to you.
The i7-7700X was a good CPU in it's day, but current CPUs can do way more work, while using much less power. If you can afford something better, it'd suggest going for that. If you can't, then the i7-700k would be good for the price.

i7700K v Ryzen 7 9700X

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Grant
Darwin NT.

To add to what Rob has said, CPDN when it comes on line again and has work which should be this week or next for Linux tasks, floating point speed is what counts. Also because on the chips there is at least on AMD ones and I think Intel as well there is only one FPU per real core, using virtual cores at least on the machines I have had that allow use of virtual cores, it actually results in a lowered throughput though some of this may be due also to limited level3 cache memory. Prime Grid on the other hand is all integer work.