This is something I'll have to deal with (on the other side as a widow in due course) in the next few years as OH has incurable cancer. We've been together 38 years, we lost our virginity together, had a family together, bought our first home together, etc. I'd not be looking for a "replacement" in any way, shape or form, as we did so much together and were genuinely soul mates. When the time comes to start "dating" again, I'll be looking for a completely different relationship, I'd be wanting to live a "different" life, not the same life with a different person. I'd want to do different things, go to different places, i.e. start a completely new chapter. Maybe even move to a different place, maybe abroad.
If I couldn't find someone to do that with, then I'd rather do it on my own rather than compromise with a new partner who I couldn't live that different life with. I certainly wouldn't want even the faintest hint of a new man trying to "compete" with what I have with my DH, I'd be clear of that from the outset. Likewise I'd not be looking for someone similar to DH. The new relationship may well lead to marriage etc, but it would be a "different" marriage, not trying to emulate the life/experiences I have with DH.
I'd not have the house (current or a future house) full of pictures of him, sentimental momentos, etc., but I'd have a respectful "presence" even in a new home with a new partner, of maybe a couple of photos (not in pride of place) and some "nod" to him with a couple of items of his on display (again, not in pride of place), all discussed and agreed with Mr New! I'd never want to forget nor replace, but wouldn't want my future to be dominated by someone who isn't here anymore either. Even if I didn't get a new partner, I'd still move on, but on my own, and likewise, would "thin out" pictures and objects of DH over time.
It's easier for me as I'm not in any way, shape or form, religious, so I'm not thinking about reuniting in heaven, or being buried together etc. I think being buried together is probably the right thing if I died shortly after DH, but once I was in a serious long term relationship with someone new, especially if married, I'd be buried with the new man. It all depends on timescales and nature of any new relationship really.
I'm also a very "head over heart" kind of person, probably ND in some way, so I think I'd be able to deal with emotional aspects of the loss better than someone who wears their heart on their sleeve. I'd be looking for a new partner who was similar, (as is DH) so I'd like to think we could start a new life without too much "emotional baggage". But if I didn't find such a person, then I'd move on on my own without a new "life partner" and hopefully fill my life with new friends, new hobbies, etc.