June 3 (Reuters) - The European Central Bank (ECB) looks set to cut rates for the first time since 2019 on Thursday, but what happens thereafter is a bigger puzzle.

Inflation has neared the bank's 2% target, but it rose more than expected in May and it remains sticky in the dominant services sector. The bloc's economy is recovering faster than expected and the jobs market remains tight, casting uncertainty on how many times the ECB will cut rates this year.

"The cut itself won't be big news. It's more the question: what's the messaging around what's to come?" said Jens Eisenschmidt, chief Europe economist at Morgan Stanley, who was previously at the ECB.

Here are five key questions for markets:

1/ Will the ECB finally cut interest rates this week?

Most likely, given just how many policymakers have all but promised a June rate cut.

The expectation is a 25 basis-point cut that will bring the ECB's deposit rate down to 3.75% from the record 4% it reached last September.

2/ What will the rates path look like after June?

That is much less certain.

Markets now expect fewer than 60 basis points of cuts this year, meaning two moves and less than a 50% chance of a third, down from three when the ECB last met in April and at least five in January.

Many forecasters still expect three cuts - in June, September and December - all meetings where the ECB releases fresh economic forecasts.

Hawks are trying to take a July move off the table. Others, like French central bank governor Francois Villeroy de Galhau, don't want to close the door.

So don't expect much guidance from ECB chief Christine Lagarde on Thursday. Analysts reckon she'll repeat the bank's mantra that it's "data dependent".

"I think they will be far less prescriptive about what comes next than they have been around the June meeting," said BNP Paribas' chief Europe economist Paul Hollingsworth.

3/ How big of a headache is accelerating wage growth for the ECB?

Not a huge one, economists reckon.

Before cutting rates, policymakers wanted to see more evidence of wage growth slowing, but data in May showed it rose back to 4.69% during the first quarter. But it was skewed by a high print from Germany, where pay growth is still catching up with inflation.

The ECB was seen as signalling it wasn't concerned, publishing a blog the same day that stressed other wage indicators point to moderating pressures. Even Germany's Joachim Nagel, a leading hawk, brushed off the data.

Yet services inflation, which reflects domestic demand, rebounded in May, while record low unemployment may also cast uncertainty on how much wages will cool.

The wage data "gives you more reason to be gradual in your rate cutting cycle, simply because you have to wait before you get more confirmation that you really will land at 2% (inflation) in time," Morgan Stanley's Eisenschmidt said.

4/ What about a strengthening euro zone economy?

That's not a cause for concern either.

The bloc's economy grew 0.3% during the first quarter, beating expectations for 0.2%. Forward-looking business activity data has also come in above forecasts, pointing to the recovery holding up.

Economists reckon the numbers are good news for the ECB. The uptick in activity could help improve weak productivity growth, blamed in part on labour hoarding, boosting confidence in slowing inflation. And growth figures are not high enough to cause concern about demand reigniting inflation.

"The latest economic data is encouraging. It takes away the argument of the most outspoken doves who say the economy is in trouble and we must cut quickly," said Reinhard Cluse, chief European economist at UBS.

5/ What will the ECB's new projections show? The bank is seen revising up its growth and inflation projections slightly, but this shouldn't derail its expectation that inflation will return to target in late 2025. "The big picture should remain the same as in March," PIMCO portfolio manager Konstantin Veit said.

(Reporting by Yoruk Bahceli and Stefano Rebaudo; editing by Amanda Cooper and Sharon Singleton)