65 yrs old, missing teeth 18 & 19 with more needing removed: Can I get partial implants?

no , that is not advisable. because in case of missing teeth, opposite teeth erupt more and start interfering normal occlusion , so restoration of missing teeth is must. best option for u is dental tourism. by that u can save upto 80% on dental treatment. dental tourism is ideal option for dental implants. by1/10 of amount u can restore yr smile.

to cover that gap where you are having those teeth extracted, you will need to get a good evaluation of bone, height and size, so you will have or either a partial or implants, those areas are very important for support and try not leave them to long without restoring it, masticatory forces , and receding gums might affect the remaining teeth.both options are good and this will depend on time and budget.

It is common to NOT replace second molars (and NEVER thirds/wisdom teeth). You certainly could do as you say, 14 and 19, and later if you feel you miss 15 and 18 you can replace them then.