Hey mama, you're juggling so much right now, respect to you!
For the subsidy question, as long as you and your husband are legally married and his income is considered in the household income calculation, you should still be eligible for the Basic Subsidy and Additional Subsidy (means-tested) — his living arrangement separately shouldn't disqualify you, but I'd recommend calling MSF or checking directly with the infant care centre you're applying to, as they can advise based on your exact situation.
For the rental flat, HDB rental allocation depends on availability so North area isn't guaranteed, but you can indicate your preference and they will try — if you do get shifted, most centres are understanding and you can request a transfer lah.
Wishing you a smooth delivery and recovery, you've got this!