When saying Lauds after the Office of Readings (nb it was done this way in monasteries originally), after the responsory of the final reading (or the Te Deum if applicable) you start the psalmody of Lauds. You conclude with the collet of Lauds, or optionally the Sunday collect of that week (which can be used for the OOR, Lauds and Vespers every day that isn’t a memorial or feast). Of course on feasts or memorials the collect for the OOR and Lauds are the same. The hymn is said at the start of the Office, and you can optionally use either the hymn for the OOR or for Lauds.
Yes. The hymn is alway said/sung. The invitatory psalm must precede the OOR if it’s the first office of the day. If you open the day with Lauds though, the invitatory verse must be said (“Lord open my lips”) but the Invitatory psalm may be omitted. You don’t say the invitatory at all if you open the day with any office other than the OOR or Lauds.
You do not say both collects and conclusions. If you combine the Office of Readings with Morning Prayer, you omit the collect and conclusion from the Office of Readings and go straight to the Psalmody of Morning Prayer and proceed from there, ending with the collect and conclusion for Morning Prayer (which, depending on the day, may or may not be what was prescribed for the Office of Readings).
And although you did not ask this, you can sing/recite the hymn for Morning Prayer at the beginning of the Office of Readings if you combine the two offices.
And finally, you always say the hymn, regardless of whether there was an invitatory psalm or not.
If you pray OR and Lauds SEPERATELY, the conclusions are said. If the OR is joined with Lauds, right after the Te Deum (or 2nd responsory) you may either sing the hymn for Lauds or omit it. If you omit, go to the Psalmody for Lauds and the rest continues as usual.
The invitatory is said at the first ‘hour’ of the day (whether OR or Lauds) If the invit. is said at the OR, you say the Deus in adiutorium to start the remaining hours
All hymns are to be said or sung during all of the hours