Quod having a baby who will just fall asleep when she is tired is the stuff dreams are made of for me! All that happens here when she gets tired is that she gets increasingly irritable until she reaches meltdown. There is no way in the world that she will acknowledge that she is tired and just close her eyes unless she is in the car or pushchair, so her sleep is entirely managed by me by looking for tired signs and then winding her down and getting her to bed. E has always followed the same pattern of set awake times between sleeps, which was around 2 hours when she was 4 months and has gradually increased over the weeks and is around 3 hours now at 8 months, but all babies are different so I think it's probably best to watch the baby rather than the clock and see if a pattern emerges.
Mother he is being a bit of a dick about it, especially as you do the lion's share while he sleeps, but I do understand his frustration. It's common sense that a baby has no concept of time or that it's the middle of the night or that they shouldn't be up at that time so there's no point losing your rag with them over it, but I can't honestly say I haven't muttered ,'for fucks sake!' under my breath at 4am when i'm getting up for the sixth time. It's fine to feel annoyed and frustrated over it but no point taking it out on the baby.
Lilly I think it's fine to sleep in a travel cot for as long as they still fit in it, but if you're at your Mum's for a while could you bring the normal cot from your house so she is a bit comfier? Could L just be unsettled over the change of sleeping location, or even that she's picked up vibes that your circumstances have changed? And did you just fancy a new name? 
Yes FX Hazle that you aren't heading into the shit sleep zone as well, although if another sleep regression is an inevitability then i'm now thinking it's better to get it out of the way?