Will the taking control by Liverpool City Region mean loss of integration with the rest of the network as far as through ticketing is concerned, and becoming like Metrolink? I certainly hope not. Currently I can purchase one ticket to travel between Birkenhead and Stafford (random example) Will this mean two separate purchases will be required in future? As another example regarding Metrolink I cannot buy a through ticket to Bury or Oldham from Liverpool, despite these being large northern towns, and previously being able to do so.
Metrolink was removed from the National Rail network because it involved a heavy rail to light rail (tram) conversion. Merseyrail is still heavy rail, so won't be removed from the NR network. As a result, through tickets to any other NR station are always going to be available, in the same way that you can still get through tickets to London Overground stations despite TfL running their own fares system.
By the way, through tickets to Metrolink are available from Liverpool. It's rarely the cheapest option, due to advance ticket options and PAYG on Metrolink, but you can for example get a Liverpool Stations to Metrolink Zone 1-4 ticket for £23.80 (undiscounted off-peak day return, with no evening restrictions) which is likely to be the simplest ticketing option to get to Bury or Oldham.
Secondly how will this affect 'City line' services which only see a train every two hours on Sundays?
Sunday frequencies being increased to weekday frequencies are fully dependent on what happens with Sunday working on Northern. Either way, it's not a cheap thing to resolve - either you hire more staff, or negotiate increased sunday working with the unions (which will require making many concessions). Probably worth it, but don't expect TfLCR or GBR to rush to solve it.
None of this in itself affects the frequency of "City line" services, which are indeed often poor (and unreliable). Presumably LCR will need to work with GBR on this but this is much more complicated because of crossed boundaries and interweaving.
Co-ordinating TfGM and TfLCR operations and ticketing will eventually be something that has to be done. It's a logistical nightmare, but the region already has so many overlapping in terms of commuting patterns that separate, unintegrated ticketing systems and railway services won't be able to last long.
The issue then becomes, where do you draw the line? As at the same time, North Wales and West Yorkshire are also developing similar systems, with some overlapping with GM and Merseyside. Every station between Rhyl and Leeds having a PAYG system, with no clear indication of how far you can actually pay as you go, will cause problems.
Looking at City Line services, I doubt much will happen unless the Liverpool-Manchester HSR goes ahead. There just isn't the capacity to run more stopping trains on the Chat Moss and CLC routes. Even if NPR/LMR/HS2 does go ahead though, old PTE-style operations where the Bee Network and City Line services run up to the boundary station (e.g. Warrington or Wigan) with zero coordination isn't a good option due to overlapping commuter patterns. For example, Widnes and Birchwood have plenty of travel to both Liverpool and Manchester.
Without proper service coordination (which GBR should also take part in) you end up with fairly uneven and inefficient routes - for example, Golborne (when it opens) could have services to both Liverpool and Manchester thanks to the western & eastern chord options near Newton-le-Willows. With a direct train, travel times to either major city would be similar. However, the station plans appear to involve 1-2tph to Manchester and absolutely nothing to Liverpool, since the station's in GM (barely). This doesn't help with Liverpool's perpetual game of catch-up against Manchester.
