There's a lot of scaremongering and emotion over this, and I hope this thread will remain civil. My own take on it is that people are good, for the most part, and it's generally wrong to judge the people of a country by the actions of their political leaders.

The facts of the situation are that people have travelled very recently in Russia without hostility from locals or officials. In fact they report that people have been at pains to welcome them as visitors to dispel the adverse notions that some foreigners have about their country. As with most Asian states there is a tradition of hospitality towards travellers, because you never know when it might be you. Of course, you can't count out the occasional racist idiot, but that's not unique to Russia.

As a rule of thumb I suggest be guided by the policy of the Russian visa authorities towards your passport. If there are no restrictions outside the normal visa rules, you should have no problems. The normal guidance applies - act like a guest, avoid discussion of politics or religion and be sensitive to local customs and activities the authorities might deem incompatible with tourist status.

Practicalities, the European Nordic states are behaving very negatively towards Russians and persons passing their borders. There are stories of Euros being confiscated and 3-4 day waits to cross. Finland has closed all its crossing points except the one in the furthest frozen north. From Europe, the Turkey-Georgia route is the best option. There are some delays due to the volume of traffic but they recently enlarged the crossing infrastructure to help handle that. Reports are that the crossing is professional and courteous.

Be aware that due to western sanctions, western ATM/credit cards, phone SIMs and medical insurance will not work in Russia. You will need to take enough currency for your stay (also allowing for emergencies). USD preferred and if you don't want to carry large amounts of folding, open a debit card account from Tinkoff, you don't need to be a resident and you can pay your Rubles into any of many ATMs and use it like a normal card. The Mir card is also accepted by some other countries in the region, which is handy if you've loaded it with more than you need. You should be able to buy a Russian SIM and road and medical insurance at the border, or if not there will be many offices in Vladikavkaz.

You won't be able to freight your bike to Russia (again, sanctions) but shipping it to a nearby neutral country (Turkey or Georgia) should present no problems. Going in to Russia you will fill out a customs form for a TIP. I forget how many days this gives you (90?) but keep to it and don't lose the TIP slip and you'll be fine.