1) Yes. Should be automatic if all on one ticket. If on separate tickets show the onward ticket at NSN check in and then it will be through checked to GLA (which is where she'll next see the checked bags).
2) Can take the free (white) transfer bus. Stop is in between NZ and QF/DJ parts of the domestic terminal - ie once outside from the prop gates turn left and walk along past several doors until she reaches the covered bridge to the carpark, the inter-terminal bus leaves from the old smokers corner in front of the terminal (the stop under the bridge is for the city bus). Buses run about every 10 minutes or so and the drive will take up to 10 minutes. Alternatively, follow the blue and white signs and walk across to international terminal, which takes about 15 minutes or so from the prop gates. Only part of the way is under shelter.
SQ check in is near the entrance closest to domestic (right in front where the inter-terminal bus drops off). For business class she can use the designated business class line (will have a blue carpet laid out and a straight line, compared with the snake for economy). Once checked in she can use the premium "other airlines" immigration desk which is against the wall closest to domestic terminal. She'll get a pre-cleared immigration sticker on her boarding pass. Then walk along to the escalators up to departure level. At the immediate left of the entrance to immigration is a line for pre-cleared immigration, which cuts out the main queue. The officer there will quickly check passport photo matches and BP and passport names match and through to security.
3) SQ uses the NZ lounge. Once through security go past the first lot of shops around the corner. The hallway to the lounges is on the right just before the zigzag. Along the corridor, turn right and up the escalator. Lounge is behind her at the top.
SQ will arrive in SIN T3. There she will have access to the Silver Kris Lounge, business class side. The lounge is not well signposted but is in the middle of T3 and up an escalator, almost opposite the stop for the airside train to T2. There is another escalator to the Kris Flyer Gold lounge - but there is no connection through to SKL from there.
SQ to MAN will depart from T3.
4) Immigration will be at MAN however customs in UK is at final airport (ie GLA). At GLA her bags will be on the special international luggage belt (ie a different belt to most of the other passengers) and if she has anything to declare she will be expected to contact customs staff (they'll either be hanging around or else contactable by phone on the wall next to the luggage belt).