Just tested again on a Samsung Galaxy S5 (previously tested OK with the stock 4.4 ROM) having just updated to Lollipop. Where I use the keyboard function directly, or click on a text field on the DroidBOX's browser and get a pop up keyboard on my phone, no double strokes or drop outs occurred.
I'm using the Swype keyboard on my phone, it might be worth experimenting with temporarily switching to a different keyboard app on your phone and tablet, see if the same issues occur. If they do, reboot modem/router (silly but worth a shot if it has been on a long time), see whether dictation is allowed by your keyboard(s), to see if the output from that comes through OK or not.
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Meant to say, can you let us know the make and model for the phone & tablet, and if you know, which version of Android is on both.