Partners Group Long Term

I know a few folks who used to work at PG in the US HQ (Colorado office) and can say a few things about the questions you asked, although this will be from a US perspective, but as the firm works globally a lot of it will still apply.

• Definitely not a bad place to work in the sense that you'll gain a lot of experience - while name brand recognition in the US isn't as good as EU, PG is still technically a MF in terms of size (although their PE directs could be considered more UMM since it is a younger group relative to other big PE direct funds in the market, and the assets are split across their 4 main arms: RE, Equity, Debt, and Infra). 
• Based on your post sounds like you got directly hired to work on the PE team as opposed to the rotational analyst hiring class, so you'll be only working with that group long term instead of the typical analyst 2-3 yearish contract where they bounce between groups hoping to get a permanent placement. I'm unclear as to how exits to other PE shops are as this is not something I have experience with, perhaps someone else can comment on that.
• Comp in US was severely under market for many years until a few years ago, when they finally increased to closer to market standard, so the gap is no longer as large, although I think they still underpay slightly relative to others in the market.
• Promotion is quite slow from what I hear, although there have been occasional cases where exceptional performers get bumped very quickly to higher levels, but quite frankly speaking that's very unlikely to happen for an analyst in PE directs and you shouldn't get your hopes up - not being mean, just realistic, as my understanding is this was in other groups for exceptional circumstances.
• Culture is quite good, at least from the perspective of a PE shop. Very collegial people who are easy to get along with, much more down-to-earth with a lot less of the "finance bro" cultures you tend to encounter on Wall Street. Flat hierarchy as well where you'll interact a lot with senior folks on the team, and you are actually encouraged to take your vacation days each year. But the firm is very, very heavy on the bureaucracy, as you mentioned. You'll still be pulling long hours as well during the major deal sprints - PE is probably going to be fairly rough. As the other commenter mentioned, try talking to some of the analysts on the team to get a sense of how things are as I myself don't have any direct exposure, only secondhand knowledge, and the culture has some variance based on location. 
• If you're based in EU my two cents is that this is probably a pretty good opportunity as my understanding is PG is quite highly regarded over there, and getting recruited directly for a PE directs role is pretty uncommon given they have a pipeline normally through the analyst program. I doubt you'll have much to gain by sticking around in EB, as buyside skillsets are much more valuable for future investment roles than banking, and firms with good cultures can be pretty tough to come by in high finance.
• As I said before, I can't really comment on exits but it seems to be quite common for analysts/associates to hop to other funds or just entirely different areas of other industries after putting in a few years. Honestly speaking, the skillset that you gain is far more important anyway as opportunities generally come from networking anyway, so I wouldn't worry too much about exit ops before you've even taken the job.