There may or may not be a vote in the House this week. Look for a Goldilocks-level of difficulty. It can't be too easy, and it can't be too difficult.
The House is never as challenging as the Senate. The Republicans have a bigger margin for error in the House, numerically, and fewer moderates, which is to say, basically none. In the Senate, they can only lose two votes, and they have Collins and Murkowski tugging them leftward. Collins and Cassidy cosponsored a more moderate bill, so count them as three collectively, and then you have the four signatories of that letter demanding a reduction of those Medicaid cuts. That's seven Senators whose positions seem less hard-line than the House.
Then, you have basic Senate rules. The Republicans are doing this through "budget reconciliation," which means that nothing included in the bill can be non-budgetary. Some of what's in the bill may have problems with the Senate parliamentarian, like the provision that insurers jack up their rates on anyone who has a lapse in coverage, since that doesn't directly touch either taxes or outlays.
The Senate is always more difficult. So, if the Republicans have too hard a time in the House this week, then things are looking really bleak for a repeal of Obamacare. So here's where things get really fun. If Ryan is smart, and things get challenging, he'll delay or cancel a vote. If his whip count comes in and he doesn't have a majority, the last thing he wants is to "get rolled" (Congress jargon for what happens when the majority party's formal position loses on the floor). That rarely happens. Why? Partially, a smart Speaker won't hold a vote if he thinks he'll get rolled. If the vote gets delayed or canceled, shit is getting bleak for Ryancare/Trumpcare/whatevercare.
But remember I said something about a Goldilocks-level of difficulty...
What about "too easy?" Well, if the vote is too easy, that means everyone knows this is theater, and they're just passing something symbolic. So, Ryan makes a bunch of concessions to the Freedom Caucus, puts the bill on the floor, they get maybe a couple of defections from, say Ileana Ros-Lehtinen and the others who pass for moderates in today's House, and call it a day, knowing that Collins, Murkowski, Cassidy and a few others will kill the whole thing in the Senate anyway, so they go home and relax.
This is only serious policy-making if it is difficult, but not too difficult. How difficult is that? I... don't know. Honestly, the Senate is the real battleground anyway, which I've been saying all along (I'm pretty sure). I'm just making two points: If they can't even manage this in the House, then they can't get it together in the Senate, and if they aren't taking it seriously in the House, they probably think it's a lost cause in the Senate.
This should be a fun week.