1) If someone is said to be ‘stringing sb along’ it can mean that they aren’t really serious about starting a relationship with that person, but, because they like the attention, or for some other reason, will do just enough to keep the other person’s attention. I’ll try and give an example -
‘We went on a date and it was ok, but sometimes she doesn’t answer my calls. Then she’ll just call me out of the blue and we’ll meet up for coffee, then I don’t hear from her again for a week. I don’t get it’
‘Sounds like she’s stringing you along’.
2. On the back burner could mean that, in this context. It’s more usual meaning is to give something a lower priority, without abandoning it altogether, but in the example you give yes it does mean what you say.