I can defo help you out with reconnaissance on those two buildings you mention for potential future urbex trips π
The synagogue is very much in use, beutiful building but I think if you wanted to look around inside it would be fine if you asked the Rabbi. But not one for sneaking into.
I know the derelict church you mention over the road. I think it might be possible to jump the fence at the side, but it is very visible from the main road. The other thing to consider with that church is that it has been derelict for as long as I can remember... and I'm nearly 40. So I reckon it would be a risky, maybe even dangerous explore especially if you went climbing around. Having said that, there is no way to be sure of that without getting a proper look inside and it has been strongly boarded up for over ten years. I've not seen inside it as far as I can remember.
There are many other sites in Liverpool tho for urbex. One interesting one would be Newsham Park Hospital, the only thing about that place is that it is used for ghost experiences now, but I think if you went during the day you'd probably be ok. Last time I rode past though they had signs up saying there were guard dogs etc.