You could visit the A50 Senator Lounge.
Until further notice, all Business Class passengers ( regardless of status ) and M&M Frequent Travelers could use the Senator Lounge's.

Correct. And just to confirm, they still actively promote this: Just the other day, the LD informed a group in front of me they could use the A50 lounge if they wanted as the A13 was crowded.

Very good as that opens earlier ;)
BTW what are typical arrival gates for LH from JFK and then LH to HAJ?

My best guesses would be Z gates for the TATL and something like A1-A19 for the onwards flight. For FRA-HAJ, they might be using a small regional jet so there's a good chance you'll be bussed to the aircraft's apron position.

LH from JFK could arrive either at B or Z, although it would most likely be a Z gate.
For your HAJ departure, it would mostly depart at low A gates as mentioned by 8mh but it might also depart from mid A or high A gates or B gates.

The LH First Class Lounge in Frankfurt Terminal 1A has opened.
I am really wondering what type of grab&go food they are offering at the moment over there or is the FCL just an area with similar grab&go offerings but a dedicated seating area for First pax & HON's with more comfortable seating.

I see this language on the LH web site about masks in the lounges:
Mouth-nose cover
Please note that in our lounges it is compulsory to wear a medical mask with the standard FFP2, KN95 and N95 or surgical masks. Everyday masks, visors and masks with valves are not permitted.

Are they actually enforcing this? If people have a cloth mask on do they get kicked out?

Yes, it is enforced. As it's the law to wear a (K)N95, FFP2 or a surgical mask by law in the airport and flights to/from Germany, you wouldn't even be able to go to the lounge entrance with a cloth mask.

Yeah, I was there too on 30/05. Also, the Air Canada and Singapore lounges are closed in LHR T2 B.

At least ZRH is now more or less ready again:

https://cimg6.ibsrv.net/gimg/www.fly...e48958c03.jpeg
https://cimg7.ibsrv.net/gimg/www.fly...75d9de618.jpeg
HON corner offer
https://cimg8.ibsrv.net/gimg/www.fly...1a5783543.jpeg
Kitchen inside SEN lounge section

The preferred security check concierge service is rather subpar at the moment, since all you get is an escort to the staff/business section of the security check, and no real priority. If you coincide with one of these buggies or staff, you still potentially wait for a longer time. On Monday, I was the first to be escorted with the new service from the First Island in A (and Maurizio was there too!), and it was packed beyond imagination. Pushing all LX flights into two slots in the morning and afternoon will undoubtedly lead to crowding in check-in, security, and gates (since Swiss also put LX flight departure gates closer together it seems).

Spoke with the head of lounges (from Gate Group) today, and they will continue to observe the demand. FCL in ZRH E is currently projected not before October 2021, and FCL in ZRH A not before July/August 2021. Limousine service will most likely take longer to resume.

Great update, thanks for sharing! Arriving tomorrow in F with connecting flight, so let's see.
Can you elude to the preferred security check concierge? Never heard or seen this (other than with ZRH Airport VIP service).

I do hope that the FCL in MUC will also open soon.

Will LH open the Lounges for "indoor" dining once the Inzidenz number is low enough? Was that 5 days under 50 (unfortunately today it was 51 after 4 days below 50)?

LH has said they will review the possibilities once the restrictions are lifted. I would not expect an immediate response but they probably have a plan they can execute on once the numbers are reliably low enough.

The quickest route to "level 2" is 5 continuous days under 50. However there is also 14 continuous days under 100, of which Frankfurt city is currently at day 10.

I am flying at the end of June. Under this scenario, would I have lounge access as it stands today?
Arrive TK from IST at 10:15am.
Depart UA to IAD at 1:15pm.
It sounds like there is a no food or beverage lounge now open in the Z gates.
I am Star Gold and in business.
Thanks.

Food & beverage is not served because of local restrictions but as you're flying out of FRA in United Polaris, you would have access to any open LH lounge at the airport except for the First Class Lounge. You should be able to access the Z50 Senator Lounge which currently offers only take-away food and limited beverages ( water, coffee, tea ). Anything regarding the offerings in late-June would be a speculation at this point so I would suggest you to check what is being offered closer to your flight date; although, your ticket and your status includes lounge access.