I really don't remember where I left off on each show....
Arrow.... end of S4? it's now on what? S7?
Flash.... beginning of S3? it on S5 now?
Time Travelers..... middle of S2? now also on S5?
Supergirl.... early S3? now on S4?
I feel like I'm missing one.....
Yeah, you're missing Black Lightning, which started not in the Arrowverse but now apparently is in it now that the shows people actually cared about are ending.
Has Batgirl/woman started?
Not yet, I believe. It's supposed to start this coming season.
Is there any intent to cross this over with DCU shows, like Titans and Doom Patrol as well?
I wouldn't imagine so, unless they took the Arrowverse shows away from The CW & put them all on WB's streaming service.
As for where I am...
The last Arrow I saw was halfway through Season 6, and they're now on Season 8.
The last Flash I saw was Season 4, and they're now on Season 6.
The last Legends of Tomorrow I saw was Season 3, and they're now on Season 5.
The last Supergirl I saw was halfway through Season 3, and they're now on Season 5. Not sure I care to ever catch up on that trainwreck of a show.
I care even less about Black Lightning than I do Supergirl.
Basically, I ignore the shows when they air and catch up on the BluRays. The recent season had a weird, staggered release with Arrow & Flash releasing last month and Supergirl/Legends releasing this month. I only have last season's Flash & Arrow sets.