I'm not sure how to tag this building, which is attached to the church. It seems to be a hall in which local events may take place, some of them non-religious.

Perhaps amenity=community_centre is appropriate? I believe this building has a similar purpose to Vine Hall (on Vine Road).

In context of overlay "Buildings" – house number 63: Residential Building – way/1279620492 via StreetComplete 57.3:

There is another address here: 63A (to the left of 63).

This is the "Molesey Veterinary Centre".
Address: 4 Park Road, East Molesey, Surrey, KT8 9LE.
Phone number: 02089791384.
Probably best to double check the phone number online, just to make sure I haven't made any errors in copying it out.
The car park in front is private, for customer use only.

via StreetComplete_ee 58.2

Possible car park here? Survey.

In context of overlay "Buildings" – way/1293046526 via StreetComplete 58.0:

This is a surgery building.
Need to check local surgeries around the area to fill in proper tag for this.
Name: The Vine Medical Centre
Phone number: 02089794200 (also double check this online).
Address: 69 Pemberton Road, East Molesey, Surrey, KT8 9LJ.
Opening hours (also maybe check this online): weekdays 8:00am - 12:45pm and 2:00pm - 6:30pm.

In context of overlay "Buildings" – Garage – way/1292896708 via StreetComplete_ee 58.2:

I mean really, it belongs to the house beside it. So merge them.

In context of overlay "Buildings" – way/1297340535 via StreetComplete_ee 58.2:

This is just part of the same house. Merge these two ways.

There is a road leading here, and some car parking space. Should be visible on imagery.

via StreetComplete_ee 58.2

Sign points to 1A, to the right of 1B and down a path here. 1A seems to be called "Mais Cottage".

via StreetComplete_ee 58.2

Possibly some kind of footpath here. Survey.