No, you can keep contributing to the back door Roth as long as you have earned income, unless the government closes the loophole (which Obama threatens to do every year but hasn't done yet.)
The 457 and 401k (probably actually a 403b if it's a government employer) have separate contribution limits, so you can contribute a total of $36,000 for next year ($18,000 to each), assuming you're under age 50. Both of these types of accounts are pre-tax, so that $36,000 gets removed from your taxable income for next year. So say you gross $140,000 from six months of fellowship and six months of your attending job during your first year out (2017). If you max out both accounts, your modified gross for that year will be $140k - $36k = $104k. This means you will not need to do a backdoor Roth next year even if single. So in 2017, you can probably just contribute directly to your Roth (an additional $5500 or possibly $6000 if the contribution amount gets raised next year), using POST-tax money. This is again separate from your two pre-tax accounts.
Your total retirement contributions for 2017 will therefore be $36k pre-tax PLUS $5500 post-tax into your Roth. Starting in 2018, when you work FT as an attending, you will need to start using the backdoor Roth. This can still be done on top of contributing the $36k to your pre-tax accounts; you just put $5500 of post-tax dollars into a traditional IRA, convert it to a Roth, and done. No taxes will be due since you already paid taxes on that money, but you also cannot take a deduction (not that you would ever qualify for a tIRA deduction anyway as an attending).
After you leave your employer, account options vary. The Roth IRA is not through your employer, so it just stays with whatever custodian you select (say, Vanguard or Schwab). After separation, you generally have a few options for the 401k or 403b: 1) roll it into your new employer's 401k/403b (if they offer that option); 2) leave it with your previous employer; 3) roll it into a traditional IRA. Option 3 is probably not a good option if you plan to do backdoor Roth contributions in the future, since you will run afoul of the pro-rata rule that WCI mentioned, and depending on your state's laws, you may also lose creditor protection for your retirement account in the event that you get sued. (That is, the 401k may be protected from creditors, but the tIRA may not be.) The 457 can similarly be rolled into another account (either new employer's retirement account or tIRA), but it may not be a good idea to do this just because the rules for accessing 457 money are different than the rules for accessing 403b/401k/IRA money. Unlike a 401k/403b/IRA, you can have access to your 457 money upon separation from your employer without an age penalty, whereas the 403b/401k/IRA money often cannot be easily accessed until age 59.5 (subject to certain exceptions).
I'm going to be starting a fellowship next summer after working for a few years as an attending. I like my current employer's 403b plans (low cost index funds, some of them institutional funds I don't have access to as an individual investor), so I plan to leave those accounts alone when I quit my job at the end of the year. I have my 457 contributions in an annuity paying 3.5%, so I am going to leave that alone also. After my fellowship, I will likely roll the 403b/457 contributions from fellowship into the accounts I currently have, or possibly roll that money into my tIRA and then convert it into my Roth if I take some time off (because the Roth conversion in this case would require me to pay taxes on that money, so I don't want to do that if I'm working as an attending).
