I have used an old German dill pickle receipe forever that doesn't use a water bath. I make brine and boil, pour over prepared jars, put on heated lids and leave them to seal on their own. These are yummy and SO crisp (because of no water bath). I have 2 that didn't seal this time. I did the usual 'push down' on the lids after they set overnight, and one of them stayed down and the other came up after several hours. Are they OK? If not, can they be kept in the refidge? How long are they good for?
In general, you shouldn't trust anything with a poor seal to last outside the fridge. In the fridge, they'll last just as long as a well-sealed jar that you've just opened. Commercially processed pickles can last a year in the fridge. I wouldn't expect homemade ones to last as long but they should still be good for long enough to eat them all, at least months!
The solution in the future is simply to switch to a different canning method, for example this lower temperature one or a boiling water bath one. Here's an example recipe - you could certainly adjust the spices to match the recipe you like, but it's of course safest to stick to that trusted recipe for the bulk ingredients.
Your problem is that this canning method doesn't reliably produce good seals. The similar inversion method, mentioned by Martin in the comments (turning the jar over after filling and closing) does work better, but even that doesn't always make a good seal! See for example this USDA source, which says:
I use a similar recipe from Cooks Illustrated for Bread and Butter Pickles. We have left them in the fridge for a very long time and have eaten them over a timeline of a year. Opening and closing the jars.
Note: I do not know what is in your brine, but mine had enough vinegar in it to keep out bacteria.