Yeah, 1% is great--almost as good as the 0% all the federal loans are at now. I refinanced mid-year last year when the expectation was the interest freeze would end around Sept/Oct, and I wanted to lock in the low rate.
Yes--interest rates rising is precisely the reason that refinance rates will go up.
You can refinance as a resident. You do need a contract typically. And you will want to be sure you're not eligible for PSLF. If you're 100% sure you won't go for PSLF, and you've got your contract, it'd be worthwhile to start your application now to lock in the low rates. You can drag out the process for a while as well--I think I was able to take a month to finish the application once the rate was locked, and then take a month to accept the final terms.
Many companies will refinance residents even without an attending contract. The rates may not be as good as if you had the attending contract in hand though, but it's possible they could be better than what they end up being in May/June. Additionally, you can always refinance your refinance...
Just be sure 100% you're not going for PSLF when you refinance--that's gone forever. So is IBR/income-based payment plans.
Laurel Road seems to be trying to become a "doctor's bank," so I think you'll often get the best rates with them, but try other as well. Use the WhiteCoatInvestor website referral link (for any refinance loan) to get an additional cash back bonus. If you are an AMA member, you get an additional 0.25% off your rate with Laurel Rd (you may have to use AMA as your referral source). For the size of my loan, it saved me a lot of money so I went ahead and became a member again--I haven't been an AMA member since med school, yet somehow they can track me to literally every one of the last 6 locations I've lived to send me multiple "dues requests." At the time there was a savings account bonus--you have to have $20,000 in their linked savings account to get the largest interest rate discount, I think it's 1.5 or 2%, so quite significant. I believe it's still offered, but you have to actually ask them about it over the phone. Otherwise the interest rate discount is based on the amount direct deposited into a linked checking account. And there's the usual autopay discount as well.
I have a physician mortgage with Laurel Rd as well, so it's one less account to deal with.