HSA limits with different tax years

You asked several questions, so let's take them one at a time.

1. "If you have a family HSA limit of $7000 for 2019 and only worked at the job for 6 months. Can you contribute only $3500 for the 2019 year total?"

The annual contribution limit is prorated by the number of months you had qualifying HDHP (High Deductible Health Plan) coverage divided by 12. So if you began work and got the HDHP coverage on July 1, then, normally, your $7,000 limit would be prorated to $3,500.

However, the IRS has a rule called the "last-month rule" This rule allows you to use the full annual HSA contribution limit, IF you had HDHP coverage on December 1, 2019 (i.e. your tax year) and did not have any disqualifying coverage like Medicare or other health coverage. So if indeed you started work on July 1 and got the HDHP coverage the same day, then you can use the full annual limit of $7,000. The only catch is that you have to stay under HDHP throughout the "testing period" (basically, all of 2020). In this case, you would report that on your 2020 return and pay extra tax.

2, "Assuming that you can only contribute the $3500 for the 2019 year if you contribute $2000 in 2019 and $1500 in January of 2020 in regular paycheck contributions. How does that $1500 affect the 2020 year? " 

You are allowed to make contributions towards your HSA for tax year 2019 up until the due date of the 2019 tax return (generally, April 15, 2020). The catch is that you MUST TELL the HSA custodian that the amount given in 2020 was for tax year 2019. Otherwise, the default behavior of the HSA custodian is to apply the contribution to the current tax year,

In this case, if you are making the HSA contributions through your employer (Payroll deduction), then you must get your employer to tell the HSA custodian that this contribution is for 2019, not 2020. Your employer may or may not be willing to do this. Usually, these contributions after the end of the year are personal contributions made by you directly to the HSA custodian where you can tell the custodian yourself which year the contribution is for.

3. "Is the technical total you can contribute for the year $7100 +$1500= $8600? Or will it be subtracted from what your limit of what you can contribute? "

The annual HSA contribution limit does not change based on when you make contributions. Because of the last-month rule (for which I assume you qualify), your contribution limit for 2019 is $7,000, whether contributed in calendar year 2019 or after the new year but before the due date of the return.

But this is why it is critical that you tell the HSA custodian which year the contribution is for, If you don't, then the $1,500 will be applied to the 2020 limit.