PPS does its lottery process for non-neighborhood schools in late winter/early spring and it is only available to current residents, so you are not going to be able to apply to ACCESS, Winterhaven, or Sunnyside Environmental for 2018-2019 if you're moving in the summer. Also, all of these schools have limited intake opportunities. ACCESS currently builds its cohort in the first and third grades, Winterhaven in kindergarten and sixth, and Sunnyside in kindergarten and sixth. The only time there are other spots is when students leave the school. Here's a PDF with the slots that were available for this year's lottery:
https://www.pps.net/cms/lib/OR01913...0MS%20Transfer%20Slots_Final_English.pdf

PPS is currently planning on converting a number of its K-8 schools in NE Portland to K-5 and creating some new 6-8 middle schools starting in fall of 2018. So you should definitely pay attention to these new maps if you're looking for housing in the near future and want to attend a certain neighborhood school. Real estate agents are not always super reliable when it comes to knowledge or disclosure of upcoming changes. Here is the summary of the proposed changes:
https://www.pps.net/cms/lib/OR01913...ddle%20School%20Transition_R10_print.pdf

OregonLive has the most recent school report cards searchable here:
http://schools.oregonlive.com/

You aren't going to find any public schools in Portland with gardening teachers. Alameda, Laurelhurst, and Beverly Cleary have the highest concentrations of TAG kids on the east side. You should probably research the pros and cons of the K-8 vs K-5 and middle school models and decide which you think would be more appropriate for your sons.