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IMHO that starting page on iBreviary app is terrible. It may be good for people who know what to do, but even then it's debatable. (Imagine trying to put on one Web page something that breviaries tend to use multiple ribbon bookmarks for; then you can get an idea of why that single-page style of organization is ill-conceived.)
For the Invitatory Psalm, you choose one of those four (95, 24, 67, 100). 95 is the most common one.
The '+' there means you cross your lips with your thumb. The hyphen there just means it's a response. (All the back and forth responses are done by yourself if you're not doing this in common.)
The reason it links to Psalmody and Hymn is because some may have already said the Invitatory (which you say once per day), and moreover the Hymn is optional in private use. So they must link to those on top in case you're aiming to skip the Invitatory sections.
The red asterisk and crosses are indicators of pauses in the line. Mainly used for chanting. If not chanting, you can safely pretend they're not there, but they do offer a place for good medial pausing.
If you're still interested in apps. You should look into the Universalis app, as it provides a better user experience, I believe. You can also set options like calendar to observe, etc. I believe this link will give you one month free use before you have to buy. It's completely offline (no network access required).
There are a few good resources out there for the LOTH, but one of the more important ones is the General Instruction of the Liturgy of the Hours. I think it is a very important read for anyone starting out (or feeling frustrated at the complexity of it all). It is important to have a good basis for further learning, and what better than the GILH, presuming you don't have a person to ask in real life?
If I missed any of your questions, let me know. You (or others) could also PM me.