I've been doing SQ11 and SQ638 every couple weeks weeks for a few months. Here's my take:

You'll arrive in T3 and depart out of T2, but that doesn't matter. It's really easy to get around.

Q1) Definately use the T3 transit hotel, its clean and fresh. The T2 lounge is horrible and smells musty (moldy?) from the humidity. You'll have to request T3 specifically, because they will try to push you into T2 given your departure gate. Better to call instead of email to confirm if you can. Also, if they say they are booked try back repeatedly, and even upon arrival if you're still not confirmed. People dont show up ALL the time (no cancelation penalty) so there is usually space avail even if they claim they're full.

Q2) No, you can't use the F lounge even if you are forced into J because its only a two class plane. SQ is really strict with their access. Access is always based on departing flight, not arriving flight.

Q3) Yes, you can use the shower in the SilverKris lounge in T2. But you can also get a private bathroom with shower in the transit hotel, that way you're all fresh to start your day.

Q4) Free internet is everywhere at Chaingi and in the lounges. You can ask the SilverKris agent to print something off for you, but it might not happen. If you check-in online and not print the boarding card, that will at least secure your seat and give you an extra 20-30 minutes.

Have a great trip!