Does your employer allow you to roll over the money? The reason I'm asking because, with $3200 in your 401(k) your account might subject to automated actions by the employer if you don't do anything soon. My employer automatically rolls over any 401(k) balance under $5000 to some Bankcorp IRA after 30 days.
Since you're asking whether you can roll over the money to accounts of your choice, yes, this is indeed the case. If your employer has policies like mentioned above, just do it soon.
Open an IRA with Vanguard once you do, Vanguard will help you with the rollover. There is usually a form you will have to send to your current IRA management.
Rollovers do not count against IRA limit. By all means, hustle up and add the max.
Now, about that new job...
you are correct that your contributions to new 401(k) account will start form 0.
Your employer's contribution does not count towards the limit.
Yes, you will be able to contribute to the IRA you open at Vanguard and to your 401(k). There are income limits on tax deductibility of contribution, however. For 2018, deductibility phases out between $63,000 and $73,000 for singles and $101,000 and $121,000 for married people filling jointly.
If your IRA contribution is not deductible, you could increase your contribution to regular investment account, which likely will be more tax efficient than nondeductible traditional IRA and not subject to required minimum distributions in the future. I hope this helps.