I've tried below to say things that I haven't already said in the other threads but sorry if I have repeated myself in places!
I'd advise against the suggestion of "having one language for each situation - Welsh at school and English at home." If you want your child to be truly bilingual then each language needs to feel natural to them and I think pigeonholing them would be counterproductive.
Also, in my experience, kids have no trouble going from one language to the other and telling you in English about their day which has been all in Welsh won't be a problem. They'll just say "What is "cath" in English?" (or whatever) and you'll look it up and say "It's 'cat'" and that is how you and they learn. Okay, that's simplified, but that's how it works, basically.
I can be on the phone to my children (now in secondary school - Welsh medium) and say "Ask Dad where he's put the cheque book" and they'll instantly say it in very natural Welsh to him, get the answer in Welsh and immediately repeat the answer to me in English. They don't repeat it word for word in the way I used to do French in school (i.e. badly!) but they use a natural spoken language. Friends of mine who both speak Welsh fluently (1st language) even say that the children speak to the mum using her North Walian words and translate (not literally, you understand, just the words that the parents wouldn't use themselves in normal speech) instantly to the dad using his South Walian words.
I'm not saying that it's going to be easy or without times of feeling like it's hard work and also you might feel that you're missing out on their schooling as you don't understand their schoolwork, but I wanted my children to be bilingual and they are and I'm so glad of that.
I'm impressed by the bit in one of the previous posts on this thread about children plotting in the language that the parent doesn't have! I remember saying to my dd that it must be great to listen in to other people's conversations and she said that the thought wouldn't enter her head! If my siblings and I had been able to speak a language that a parent couldn't, I'm sure we would've taken advantage and listened into conversations! My children don't speak Welsh with each other naturally as English is their first language being my only fluent one, but that's understandable. It would be lovely if they did, though.
Sorry to waffle. If you need any advice or support and you are able to, please CAT me if you want to and I can give you details of organisations which are set up to help English speaking parents whose children are being schooled through the medium of Welsh.