3m Lead Test negative for days then turned positive

It seems that this document answers most of your questions:

Testing for Lead

In the document it states that home kits are available, but it has been shown that the effectiveness is not too reliable, and if you do detect lead, you will probably need to get soil professionally analysed.

otherwise you can find do-it yourself test kits on this link if needed. DIY Test kit

Personally, I would inspect the soil for any traces of paint chips and if you find any, then get lab tests. however a sensible course of action would be to order in some top-soil for your garden, that way, your garden will look nice in a few weeks and you will have peace of mind in terms of contaminants.

Lead isn't soluble in water (although some lead containing compounds are), so most filtration smaller than ~1 micron is typically quite effective at removing lead. Just look for a filter with a NSF/ANSI Standard 53 rating - it needs a 99% lead reduction to certify (link leads to a specific filter, but does list the qualifying criteria).

Other contaminants can actually be much more difficult to remove - I'd assume if any lead even made it to the carbon block, that vast majority of it would stop there.